i-  J . 


eT- 


ERNHARP  FRAN7., 
42  South,  0  Street, 

Virginia,  AT.  7". 

seller,  f»tatlon«»r 


2jmsic  a«a  FAycr  WOPS.  ., 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

OP" 


Received 
Accession  No. 


Class  No. 


\ 


THE 


PRACTICAL 

MODEL  CALCULATOR, 


FOB   THE 


ENGINEER,   MECHANIC,   MACHINIST, 

MANUFACTURER  OF  ENGINE-WORK,  NAVAL  ARCHITECT, 
MINER,  AND   MILLWRIGHT. 


BY 

OLIVER  BYRNE, 

CIVIL,     MILITARY,     AND    , MECHANICAL     ENGINEEE. 

Compiler  and  Editor  of  the  "Dictionary  of  Machines,  Mechanics,  Engine-work,  and  Engineering-" 

Author  of  "  The  Companion  for  Machinists,  Mechanics,  and  Engineers ;"  Author  and  Inventor 

of  a  New  Science,  termed  "  The  Calculus  of  Form,"  a  substitute,  fur  the.  differential 

and  Integral  Calculus;  "  The  Elements  of  Euclid  by  Colours"  and  numerous 

otfter  Mathematical  and  Mechanical  Works.    Surveyor-General  of  the 

English  Settlements  in  the  Falkland  Isles.     Professor  of 

Mathematics,  College  of  Civil  Engineers,  London. 


UNIVERSITY 
CALIFO? 


PHILADELPHIA: 

HENRY     CAREY     B  A  I  R  D, 

406    WALNUT    STREET. 

1863. 


6  THE   PRACTICAL  MODEL   CALCULATOR. 

The  Litre  for  liquid  measure  is  a  cubic  decimetre  =  1-76077 
imperial  pints  English,  at  the  temperature  of  melting  ice ;  a  litre 
of  distilled  water  weighs  15434  grains  troy. 

The  unit  of  weight  is  the  gramme :  it  is  the  weight  of  a  cubic 
centimetre  of  distilled  water,  or  of  a  millilitre,  and  therefore  equal 
to  15-434  grains  troy. 

The  kilogramme  is  the  weight  of  a  cubic  decimetre  of  distilled 
water,  at  the  temperature  of  maximum  density,  4°  centigrade. 
The  pound  troy  contains  5760  grains. 
The  pound  avoirdupois  contains  7000  grains. 
The  English  imperial  gallon  contains  277*274  cubic  inches  ;  and 
the  English  corn  bushel  contains  eight  such  gallons,  or  2218-192 
cubic  inches. 

APOTHECARIES'  WEIGHT. 

Grains marked gr. 

20  Grains  make  1  Scruple      —    sc.  or  9 

3  Scruples  —    1  Dram         —    dr.  or  3 

8  Drams    —     1  Ounce        —    oz.  or  3 

12  Ounces   —     1  Pound        —    Ib.  or  ft>. 

fr.         sc. 
0=1       dr. 
60  =      3=1       oz. 
480  =    24  =    8  =    1      Ib. 
5760  =  288  =  96  =  12  =  1 

This  is  the  same  as  troy  weight,  only  having  some  different 
divisions.  Apothecaries  make  use  of  this  weight  in  compounding 
their  medicines ;  but  they  buy  and  sell  their  drugs  by  avoirdupois 
weight. 

AVOIRDUPOIS  WEIGHT. 

Drams marked  dr. 

16  Drams make  1  Ounce —      oz. 

16  Ounces —    1  Pound —      Ib. 

28  Pounds —    1  Quarter —       qr. 

4  Quarters  —    1  Hundred  Weight. . .    —       cwt. 

20  Hundred  Weight...    —    1  Ton —      ton. 

dr.  oz. 

16  =          1  Ib. 

256  =        16  =        1        qr. 
7168  =      448  =      28  =    1       cwt. 
28672  =    1792  =    112  =    4  =    1      ton. 
573440  =  35840  =  2240  =  80  =  20  =  1 

By  this  weight  are  weighed  all  things  of  a  coarse  or  drossy 
nature,  as  Corn,  Bread,  Butter,  Cheese,  Flesh,  Grocery  Wares, 
and  some  Liquids ;  also  all  Metals  except  Silver  and  Gold. 

Oz.  Dwt.  Gr. 

Note,  that  1  Ib.  avoirdupois  =  14     11     15J  troy. 
1  oz.         —         =    0     18       5J   — 
Idr.        —         =    0       1       31  — 


WEIGHTS   AND   MEASURES. 


TROY   WEIGHT. 


Grains marked  Gr. 

24  Grains  make  1  Pennyweight  Dwt. 
20  Pennyweights  1  Ounce          Oz. 
12  Ounces  1  Pound         Lb. 


Gr.     Dwt. 
24  =      1      Oz. 
480  =    20  =    1     Lb. 
5760  =  240  =  12  =  1 


By  this  weight  are  weighed  Gold,  Silver,  and  Jewels. 

LONG   MEASUEE. 


12    Inches  

.  —    1  Foot  

.     —     Ft. 

3    Feet  

.  _     1  Yard  

.  ..    —     Yd. 

6    Feet  

.  —     1  Fathom  

_.      Fth. 

5    Yards  and  a  half  

.  —     1  Pole  or  Rod 

...     -—      PI 

40    Poles  

.  —     1  Furlong  . 

—      Fur 

8    Furlongs  

.  —     1  Mile  

—     Mile. 

3    Miles  

.  —     1  Leasrue  .  . 

.....     —      Lea. 

69g  Miles  nearly  

.  -  —     1  Degree  

—     Desr,  or 

In.          Ft. 
12  =        1 

36  =        3 

198  =      16J 
7920  =    660 
63360  =  5280 

Yd. 
=        1          PI. 

-        51=      1 
=    220    =    40  = 
=  1760    =  320  = 

Fur. 
1       Mile. 
8    =   1 

CLOTH   MEASURE. 

2  Inches  and  a  quarter.... make  1  Nail marked  Nl. 

4  Nails —     1  Quarter  of  a  Yard..     —      Qr. 

3  Quarters , —     1  Ell  Flemish —      E  F. 

4  Quarters —     1  Yard —      Yd. 

5  Quarters —     1  Ell  English —      EE. 

4Qrs.lJInch —    1  Ell  Scotch —      E  S-. 

SQUARE   MEASURE. 

144    Square  Inches make  1  Sq.  Foot marked  Ft. 

9    Square  Feet —     1  Sq.  Yard —      Yd. 

30J  Square  Yards —     1  Sq.  Pole —      Pole. 

40    Square  Poles '..'—-     1  Rood —      Rd. 

4    Roods —     1  Acre —      Acr. 

Sq.  Inc.       Sq.  Ft. 

144=          1        Sq.  Yd. 
1296  =          9    =        1        Sq.  PL 
39204  =      272J  =      30J  =      1       Rd. 
1568160  =  10890    =  1210    =    40  =  1      Acr. 
6272640  =  43560    =  4840    =  160  =  4  =  1 

When  three  dimensions  are  concerned,  namely,  length,  breadth, 
and  depth  or  thickness,  it  is  called  cubic  or  solid  measure,  which  is 
used  to  measure  Timber,  Stone,  &c. 

The  cubic  or  solid  Foot,  which  is  12  inches  in  length,  and  breadth, 
and  thickness,  contains  1728  cubic  or  solid  inches,  and  27  solid 
feet  make  one  solid  yard. 


THE  PRACTICAL  MODEL  CALCULATOR. 


DRY,   OR   CORN   MEASURE. 

2  Pints  

.  .  .make 

1  Quart  marked 

Qt. 

2  Quarts  

...  — 

1  Pottle  — 

Pot. 

2  Pottles  

...  — 

1  Gallon  — 

Gal. 

2  Gallons  

...  — 

IPeck  — 

Pec. 

4  Pecks  

...   — 

1  Bushel  — 

Bu. 

8  Bushels  

...   — 

1  Quarter  — 

Qr. 

5  Quarters  — 

,..   — 

1  Weigh  or  Load...     — 

Wey. 

2  Weys  

...  — 

1  Last  — 

Last. 

Pts. 

Gal. 

,    8  = 

1 

Pec. 

16  = 

2  = 

1 

Bu. 

64  = 

8  = 

4  = 

1       Qr. 

512  = 

64  = 

32  = 

8=1      Wey. 

2560  = 

320  = 

160  = 

40  =    5  =  1      Last. 

5120  = 

640  = 

320  = 

80  =  10  =  2  =  1 

WINE 

MEASURE. 

2  Pints  

...make 

1  Quart  marked 

Qt. 

2  Quarts  , 

..   — 

1  Gallon  — 

Gal. 

42  Gallons.... 

..  — 

1  Tierce  — 

Tier. 

63  Gallons  or 

IJ  Tier..  — 

1  Hogshead  — 

Hhd. 

2  Tierces...., 

..  — 

1  Puncheon  — 

Pun. 

2  Hogsheads 

..   — 

1  Pipe  or  Butt  — 

Pi. 

2  Pipes  , 

,..  — 

ITun  — 

Tun. 

Pts.  Qts. 
2  =   1   Gal. 
8  =   4=1  Tier. 

336  =  168  =  42  =  1   Hhd. 

504  =  252  =  63  =  1J  =  1   Pun. 

672  =  336  =  84  =  2  =  1}  =  1   Pi. 

1008  =  504  =  126  =  3  =2  =  1J  =  1  Tun. 
2016  =  1008  =  252  =  6  =4  =3  =2  =  1. 

ALE  AND  BEER  MEASURE. 

2  Pints make  1  Quart marked  Qt. 


4  Quarts 

36  Gallons 

1  Barrel  and  a  half. . . . 

2  Barrels 

2  Hogsheads 

2  Butts.., 


—  1  Gallon.... 

—  1  Barrel 

—  1  Hogshead. 

—  1  Puncheon. 

—  IButt 

—  ITun.., 


Gal. 

Bar. 

Hhd. 

Pun. 

Butt. 

Tun. 


Pts.        Qt. 

2  =      1        Gal. 

8  =      4  =      1      Bar. 
288  =  144  =    36  =  1      Hhd. 
432  =  216  =    54  =  1J  =  1      Butt. 
864  =  432  =  108  =  3    =2  =  1 


OF   TIME.  9 

OF  TIME, 

60  Seconds make  1  Minute marked  M.  or'. 

60  Minutes —    1  Hour —       Hr. 

24  Hours —    1  Day —      Day. 

7  Days —    IWeek —      Wk. 

4  Weeks —    1  Month —       Mo. 

13MonthMDay,6 Hours,  j  _    1  j  H      y  __      y 

or  365  Days,  6  Hours.      / 
Sec.  Min. 

60  =  1          Hr. 

3600  =          60  =        1       Day. 
86400  =      1440  =      24  =      1       Wk. 
604800  =    10080  =    168  =      7=1      Mo. 
2419200  =    40320  =    672  =    28    =4  =  1 
31557600  =  525960  =  8766  =  365J  =  1  Year. 

Wk.Da.Hr.    Mo.  Da.  Hr. 
Or  52     1     6  =  13     1     6  =  1  Julian  Year. 

Da.  Hr.  M.   Sec. 
But  365     5     48     48  =  1  Solar  Year. 

The  time  of  rotation  of  the  earth  on  its  axis  is  called  a  sidereal 
day,  for  the  following  reason :  If  a  permanent  object  be  placed  on 
the  surface  of  the  earth,  always  retaining  the  same  position,  it  may 
be  so  located  as  to  be  posited  in  the  same  plane  with  the  observer 
and  some  selected  fixed  star  at  the  same  instant  of  time ;  although 
this  coincidence  may  be  but  momentary,  still  this  coincidence  con- 
tinually recurs,  and  the  interval  elapsed  between  two  consecutive 
coincidences  has  always  throughout  all  ages  appeared  the  same. 

It  is  this  interval  that  is  called  a  sidereal  day. 

The  sidereal  day  increased  in  a  certain  ratio,  and  called  the 
mean  solar  day,  has  been  adopted  as  the  standard  of  time. 

Thus,  366-256365160  sidereal  days  =  366-256365160  -  1  or 
365-256365160  mean  solar  days,  whence  sidereal  day  :  mean  solar 
day  : :  365-256365160  :  366-256365160  : :  0-997269672  :  1  or  as 
1  :  1-002737803,  when  23  hours,  56  minutes  4-0996608  sec.  of 
mean  solar  time  =  1  sidereal  day;  and  24  hours,  3  minutes, 
56-5461797  sec.  of  sidereal  time  =  1  mean  solar  day. 

The  true  solar  day  is  the  interval  between  two  successive  coinci- 
dences of  the  sun  with  a  fixed  object  on  the  earth's  surface,  bring- 
ing the  sun,  the  fixed  object,  and  the  observer  in  the  same  plane. 

This  interval  is  variable,  but  is  susceptible  of  a  maximum  and 
minimum,  and  oscillates  about  that  mean  period  which  is  called  a 
mean  solar  day. 

Apparent  or  true  time  is  that  which  is  denoted  by  the  sun-dial, 
from  the  apparent  motion  of  the  sun  in  its  diurnal  revolution,  and 
differs  several  minutes  in  certain  parts  of  the  ecliptic  from  the 
mean  time,  or  that  shown  by  the  clock.  The  difference  is  called 
the  equation  of  time,  and  is  set'  down  in  the  almanac,  in  order  to 
ascertain  the  true  time. 


ARITHMETIC. 


ARITHMETIC  is  the  art  or  science  of  numbering;  being  that 
branch  of  Mathematics  which  treats  of  the  nature  and  properties 
of  numbers.  When  it  treats  of  whole  numbers,  it  is  called  Com- 
mon Arithmetic ;  but  when  of  broken  numbers,  or  parts  of  num- 
bers, it  is  called  Fractions. 

Unity,  or  a  Unit,  is  that  by*  which  every  thing  is  called  one ; 
being  the  beginning  of  number ;  as  one  man,  one  ball,  one  gun. 

Number  is  either  simply  one,  or  a  compound  of  several  units ; 
as  one  man,  three  men,  ten  men. 

An  Integer  or  Whole  Number,  is  some  certain  precise  quantity 
of  units ;  as  one,  three,  ten.  These  are  so  called  as  distinguished 
from  Fractions,  which  are  broken  numbers,  or  parts  of  numbers ; 
as  one-half,  two-thirds,  or  three-fourths. 

NOTATION  AND  NUMERATION. 

NOTATION,  or  NUMERATION,  teaches  to  denote  or  express  any  pro- 
posed number,  either  by  words  or  characters  ;  or  to  read  and  write 
down  any  sum  or  number. 

The  numbers  in  Arithmetic  are  expressed  by  the  following  ten 
digits,  or  Arabic  numeral  figures,  which  were  introduced  into 
Europe  by  the  Moors  about  eight  or  nine  hundred  years  since : 
viz.  1  one,  2  two,  3  three,  4  four,  5  five,  6  six,  7  seven,  8  eight, 
9  nine,  0  cipher  or  nothing.  These  characters  or  figures  were 
formerly  all  called  by  the  general  name  of  Ciphers;  whence  it 
came  to  pass  that  the  art  of  Arithmetic  was  then  often  called 
Ciphering.  Also,  the  first  nine  are  called  Significant  Figures,  as 
distinguished  from  the  cipher,  which  is  quite  insignificant  of  itself. 

Besides  this  value  of  those  figures,  they  have  also  another,  which 
depends  upon  the  place  they  stand  in  when  joined  together ;  as  in 
the  following  Table: 


*   1   I 

3         §         a 


d       §     I     i     1      1     .a      * 
««       w     £      s     w      H     H      w 

&c.      9 


8 

7 

6 

5 

4 

3 

2 

1 

9 

8 

7 

6 

5 

4 

3 

2 

9 

8 

7 

6 

5 

4 

3 

9 

8 

7 

6 

5 

4 

9 

8 

7 

6 

5 

9 

8 

7 

6 

9 

8 

7 

9 

8 

9 

10 


NOTATION  AND   NUMERATION.  11 

Here  any  figure  in  the  first  place,  reckoning  from  right  to  left, 
denotes  only  its  own  simple  value ;  but  that  in  the  second  place 
denotes  ten  times  its  simple  value ;  and  that  in  the  third  place 
a  hundred  times  its  simple  value ;  and  so  on ;  the  value  of  any 
figure,  in  each  successive  place,  being  always  ten  times  its  former- 
value. 

Thus,  in  the  number  1796,  the  6  in  the  first  place  denotes  only 
six  units,  or  simply  six ;  9  in  the  second  place  signifies  nine  tens, 
or  ninety ;  7  in  the  third  place,  seven  hundred ;  and  the  1  in  the 
fourth  place,  one  thousand ;  so  that  the  whole  number  is  read  thus — 
one  thousand  seven  hundred  and  ninety-six. 

As  to  the  cipher  0,  it  stands  for  nothing  of  itself,  but  being 
joined  on  the  right-hand  side  to  other  figures,  it  increases  their 
value  in  the  same  tenfold  proportion :  thus,  5  signifies  only  five ; 
but  50  denotes  5  tens,  or  fifty;  and  500  is  five  hundred;  and 
so  on. 

For  the  more  easily  reading  of  large  numbers,  they  are  divided 
into  periods  and  half-periods,  each  half-period  consisting  of  three 
figures ;  the  name  of  the  first  period  being  units ;  of  the  second, 
millions ;  of  the  third,  millions  of  millions,  or  bi-millions,  contracted 
to  billions ;  of  the  fourth,  millions  of  millions  of  millions,  or  tri- 
millions,  contracted  to  trillions ;  and  so  on.  Also,  the  first  part 
of  any  period  is  so  many  units  of  it,  and  the  latter  part  so  many 
thousands. 

The  following  Table  contains  a  summary  of  the  whole  doc- 
trine : 


Periods. 

Quadrill.; 

Trillions  ; 

Billions  ; 

Millions  ; 

Units. 

Half-per. 

th.    un. 

th.   un. 

th.   un. 

th.   un. 

th.   un. 

Figures. 

123,456; 

789,098; 

765,432; 

101,234; 

567,890. 

NUMERATION  is  the  reading  of  any  number  in  words  that  is  pro- 
posed or  set  down  in  figures. 

NOTATION  is  the  setting  down  in  figures  any  number  proposed  in 
words. 

OF  THE  EOMAK  NOTATION. 

The  Romans,  like  several  other  nations,  expressed  their  numbers 
by  certain  letters  of  the  alphabet.  The  Romans  only  used  seven 
numeral  letters,  being  the  seven  following  capitals  :  viz.  I  for  one  ; 
V  for  five  ;  X  for  ten  ;  L  for  fifty  ;  C  for  a  hundred  ;  D  for  five  hun- 
dred ;  M  for  a  thousand.  The  other  numbers  they  expressed  by 
various  repetitions  and  combinations  of  these,  after  the  following 
manner : 


12  THE   PKACTICAL   MODEL   CALCULATOR. 

1=1. 

2  =  II.  As  often  as  any  character  is  repeated, 

3  =  III.  so  many  times  is  its  value  repeated. 

4  =  IIII.  or  IV.          A   less  character   before   a  greater 

5  =  V.  diminishes  its  value. 

6  =  VI.  A  less  character  after  a  greater  in- 

7  =  VII.  creases  its  value. 

8  =  VIII. 

9  =  IX. 
10  =  X. 
50  =  L. 

100  =  C. 

500  =  D  or  10.  j?or  eyery  Q  annexed,  this  becomes 

ten  times  as  many. 

1000  =  M  or  CIO.  For  every  C  and  0,  placed  one  at  each 

2000  =  MM.  end,  it  becomes  ten  times  as  much. 

5000  =  V  or  100.  A  bar  over  any  number  increases  it 

6000  =  VI.  1000  fold. 

10000  =^X  or  CCIOO. 
50000  =  L  or  1000. 
60000  =  LX. 
100000  =  <3_or  CCCIOOO. 
1000000  =  Mor  CCCCIOOOO. 
2000000  =  MM. 
&c.  &c. 

EXPLANATION   OF   CERTAIN   CHARACTERS. 

There  are  various  characters  or  marks  used  in  Arithmetic  and 
Algebra,  to  denote  several  of  the  operations  and  propositions ;  the 
chief  of  which  are  as  follow : 


4-  signifies  plus,  or  addition. 


minus,  or  subtraction. 


X  multiplication. 


division. 


:  ::  :  proportion. 

= equality. 

V square  root. 

•ty cube  root,  &c. 


Thus, 

5  +  3,  denotes  that  3  is  to  be  added  to  5  =  8. 

6  —  2,  denotes  that  2  is  to  be  taken  from  6  =  4. 
7x3,  denotes  that  7  is  to  be  multiplied  by  3  =  21. 
8  -5-  4,  denotes  that  8  is  to  be  divided  by  4  =  2. 

2  :  3  ::  4  :  6,  shows  that  2  is  to  3  as  4  is  to  6,  and  thus,  2x6=3x4. 
6  +  4  =  10,  shows  that  the  sum  of  6  and  4  is  equal  to  10. 

</3,  or  3*,  denotes  the  square  root  of  the  number  3  =  1-7320508. 

^5,  or  5*,  denotes  the  cube  root  of  the  number  5  =  1-709976. 
72,  denotes  that  the  number  7  is  to  be  squared  =  49. 
83,  denotes  that  the  number  8  is  to  be  cubed  =  512. 
&c. 


RULE   OF  THEEE.  13 

RULE  OF  THREE. 

THE  RULE  OF  THREE  teaches  how  to  find  a  fourth  proportional 
to  three  numbers  given.  Whence  it  is  also  sometimes  called  the 
Rule  of  Proportion.  It  is  called  the  Rule  of  Three,  because  three 
terms  or  numbers  are  given  to  find  the  fourth ;  and  because  of  its 
great  and  extensive  usefulness,  it  is  often  called  the  Golden  Rule. 

This  Rule  is  usually  considered  as  of  two  kinds,  namely,  Direct 
and  Inverse. 

The  Rule  of  Three  Direct  is  that  in  which  more  requires  more,  or 
less  requires  less.  As  in  this :  if  3  men  dig  21  yards  of  trench  in 
a  certain  time,  how  much  will  6  men  dig  in  the  same  time  ?  Here 
more  requires  more,  that  is,  6  men,  which  are  more  than  3  men, 
will  also  perform  more  work  in  the  same  time.  Or  when  it  is  thus : 
if  6  men  dig  42  yards,  how  much  will  3  men  dig  in  the  same  time  ? 
Here,  then,  less  requires  less,  or  3  men  will  perform  proportionally 
less  work  than  6  men  in  the  same  time.  In  both  these  cases,  then, 
the  Rule,  or  the  Proportion,  is  Direct ;  and  the  stating  must  be 

thus,  As  3  :  21  :  :  6  :  42, 
or  thus,  As  6  :  42  : :  3  :  21. 

But,  the  Rule  of  Three  Inverse  is  when  more  requires  less,  or 
less  requires  more.  As  in  this :  if  3  men  dig  a  certain  quantity 
of  trench  in  14  hours,  in  how  many  hours  will  6  men  dig  the  like 
quantity?  Here  it  is  evident  that  6  men,  being  more  than  3,  will 
perform  an  equal  quantity  of  work  in  less  time,  or  fewer  hours. 
Or  thus :  if  6  men  perform  a  certain  quantity  of  work  in  7  hours, 
in  how  many  hours  will  3  men  perform  the  same?  Here  less 
requires  more,  for  3  men  will  take  more  hours  than  6  to  perform 
the  same  work.  In  both  these  cases,  then,  the  Rule,  or  the  Pro- 
portion, is  Inverse  ;  and  the  stating  must  be 

thus,  As  6  :  14  : :  3  :    7, 
or  thus,  As  3  :    7  :  :  6  :  14. 

And  in  all  these  statings  the  fourth  term  is  found,  by  multiply- 
ing the  2d  and  3d  terms  together,  and  dividing  the  product  by  the 
1st  term. 

Of  the  three  given  numbers,  two  of  them  contain  the  supposi- 
tion, and  the  third  a  demand.  And  for  stating  and  working  ques- 
tions of  these  kinds  observe  the  following  general  Rule : 

RULE. — State  the  question  by  setting  down  in  a  straight  line  the 
three  given  numbers,  in  the  following  manner,  viz.  so  that  the  2d 
term  be  that  number  of  supposition  which  is  of  the  same  kind  that 
the  answer  or  4th  term  is  to  be ;  making  the  other  number  of  sup- 
position the  1st  term,  and  the  demanding  number  the  3d  term, 
when  the  question  is  in  direct  proportion;  but  contrariwise,  the 
other  number  of  supposition  the  third  term,  and  the  demanding 
number  the  1st  term,  when  the  question  has  inverse  proportion. 

Then,  in  both  cases,  multiply  the  2d  and  3d  terms  together,  and 
divide  the  product  by  the  first,  which  will  give  the  answer,  or  4th 
term  sought,  of  the  same  denomination  as  the  second  term. 

B 


14  THE  PRACTICAL  MODEL  CALCULATOR. 

Note,  If  the  first  and  third  terms  consist  of  different  denomina- 
tions, reduce  them  both  to  the  same ;  and  if  the  second  term  be  a 
compound  number,  it  is  mostly  convenient  to  reduce  it  to  the  lowest 
denomination  mentioned.  If,  after  division,  there  be  any  remainder, 
reduce  it  to  the  next  lower  denomination,  and  divide  by  the  same 
divisor  as  before,  and  the  quotient  will  be  of  this  last  denomina- 
tion. Proceed  in  the  same  manner  with  all  the  remainders,  till 
they  be  reduced  to  the  lowest  denomination  which  the  second  term 
admits  of,  and  the  several  quotients  taken  together  will  be  the 
answer  required. 

Note  also,  The  reason  for  the  foregoing  Rules  will  appear  when 
we  come  to  treat  of  the  nature  of  Proportions.  Sometimes  also 
two  or  more  statings  are  necessary,  which  may  always  be  known 
from  the  nature  of  the  question. 

An  engineer  having  raised  100  yards  of  a  certain  work  in 
24  days  with  5  men,  how  many  men  must  he  employ  to  finish  a 
like  quantity  of  work  in  15  days  ? 

da.  men.      da.  men. 
As  15  :  5  :  :  24  :  8  Ans. 
5 

15)120(8  Answer. 
120 


COMPOUND  PROPORTION. 

COMPOUND  PROPORTION  teaches  how  to  resolve  such  questions  as 
require  two  or  more  statings  by  Simple  Proportion;  and  that, 
whether  they  be  Direct  or  Inverse. 

In  these  questions,  there  is  always  given  an  odd  number  of  terms, 
either  five,  or  seven,  or  nine,  &c.  These  are  distinguished  into 
terms  of  supposition  and  terms  of  demand,  there  being  always  one 
term  more  of  the  former  than  of  the  latter,  which  is  of  the  same 
kind  with  the  answer  sought. 

RULE. — Set  down  in  the  middle  place  that  term  of  supposition 
which  is  of  the  same  kind  with  the  answer  sought.  Take  one  of 
the  other  terms  of  supposition,  and  one  of  the  demanding  terms 
which  is  of  the  same  kind  with  it;  then  place  one  of  them  for  a 
first  term,  an<J.  the  other  for  a  third,  according  to  the  directions 
given  in  the  Rule  of  Three.  Do  the  same  with  another  term  of 
supposition,  and  its  corresponding  demanding  term ;  and  so  on  if 
there  be  more  terms  of  each  kind ;  setting  the  numbers  under  each 
other  which  fall  all  on  the  left-hand  side  of  the  middle  term,  and 
the  same  for  the  others  on  the  right-hand  side.  Then  to  work. 

By  several  Operations. — Take  the  two  upper  terms  and  the  mid- 
dle term,  in  the  same  order  as  they  stand,  for  the  first  Rule  of 
Three  question  to  be  worked,  whence  will  be  found  a  fourth  term. 
Then  take  this  fourth  number,  so  found,  for  the  middle  term  of  a 
second  Rule  of  Three  question,  and  the  next  two  under  terms  in  the 
general  stating,  in  the  same  order  as  they  stand,  finding  a  fourth 


OF   COMMON   FRACTIONS.  15 

term  from  them ;  and  so  on,  as  far  as  there  are  any  numbers  in  the 
general  stating,  making  always  the  fourth  number  resulting  from 
each  simple  stating  to  be  the  second  term  of  the  next  following  one. 
So  shall  the  last  resulting  number  be  the  answer  to  the  question. 

By  one  Operation. — Multiply  together  all  the  terms  standing 
under  each  other,  on  the  left-hand  side  of  the  middle  term ;  and,  in 
like  manner,  multiply  together  all  those  on  the  right-hand  side  of 
it.  Then  multiply  the  middle  term  by  the  latter  product,  and 
divide  the  result  by  the  former  product,  so  shall  the  quotient  be 
the  answer  sought. 

How  many  men  can  complete  a  trench  of  135  yards  long  in 
8  days,  when  16  men  can  dig  54  yards  in  6  days  ? 

General  stating. 
yds.    54  :  16  men  : :  135  yds. 
days     8  6  days 

432  810 

16 

4860 
810 

432 )  12960  ( 30  Ans.  by  one  operation. 
1296 
0 


The  same  by  two  operations. 


1st. 


As  54  :  16  : :  135  :  40 

16 
810 
135 


54)2160(40 
216 


0 


2d. 


As  8  :  40  :  :  6  :  30 


__ 
8)240(30  Ans. 


24 


0 


OF  COMMON  FRACTIONS. 

A  FRACTION,  or  broken  number,  is  an  expression  of  a  part,  or 
some  parts,  of  something  considered  as  a  whole. 

It  is  denoted  by  two  numbers,  placed  one  below  the  other,  with 
a  line  between  them : 

,        3  numerator      ) 
U3'Tdenominator  }  which  is  named  t^-fourths. 

The  Denominator,  or  number  placed  below  the  line,  shows  how 
many  equal  parts  the  whole  quantity  is  divided  into ;  and  repre- 
sents the  Divisor  in  Division.  And  the  Numerator,  or  number  set 
above  the  line,  shows  how  many  of  those  parts  are  expressed  by  the 
Fraction;  being  the  remainder  after  division.  Also,  both  these 
numbers  are,  in  general,  named  the  Terms  of  the  Fractions. 


16  THE  PRACTICAL  MODEL  CALCULATOR. 

Fractions  are  either  Proper,  Improper,  Simple,  Compound,  or 
Mixed. 

A  Proper  Fraction  is  when  the  numerator  is  less  than  the  deno- 
minator; as  J,  or  f,  or  f,  &c. 

An  Improper  Fraction  is  when  the  numerator  is  equal  to,  or 
exceeds,  the  denominator;  as  |,  or  f,  or  f,  &c. 

A  Simple  Fraction  is  a  single  expression  denoting  any  number 
of  parts  of  the  integer ;  as  f ,  or  |. 

A  Compound  Fraction  is  the  fraction  of  a  fraction,  or  several 
fractions  connected  with  the  word  of  between  them ;  as  \  of  f ,  or 
§  of  |  of  3,  &c. 

A  Mixed  Number  is  composed  of  a  whole  number  and  a  fraction 
together ;  as  3J,  or  12f ,  &c. 

A  whole  or  integer  number  may  be  expressed  like  a  fraction,  by 
writing  1  below  it,  as  a  denominator ;  so  3  is  f ,  or  4  is  \ ,  &c. 

A  fraction  denotes  division ;  and  its  value  is  equal  to  the  quo- 
tient obtained  by  dividing  the  numerator  by  the  denominator ; 
so  y  is  equal  to  3,  and  \°  is  equal  to  4. 

Hence,  then,  if  the  numerator  be  less  than  the  denominator,  the 
value  of  the  fraction  is  less  than  1.  If  the  numerator  be  the  same 
as  the  denominator,  the  fraction  is  just  equal  to  1.  And  if  the 
numerator  be  greater  than  the  denominator,  the  fraction  is  greater 
than  1. 

KEDUCTION   OF   FRACTIONS. 

REDUCTION  OF  FRACTIONS  is  the  bringing  them  out  of  one  form  or 
denomination  into  another,  commonly  to  prepare  them  for  the  opera- 
tions of  Addition,  Subtraction,  &c.,  of  which  there  are  several  cases. 

To  find  the  greatest  common  measure  of  two  or  more  numbers. 

The  Common  Measure  of  two  or  more  numbers  is  that  number 
which  will  divide  them  both  without  a  remainder :  so  3  is  a  com- 
mon measure  of  18  and  24 ;  the  quotient  of  the  former  being  6, 
and  of  the  latter  8.  And  the  greatest  number  that  will  do  this, 
is  the  greatest  common  measure :  so  6  is  the  greatest  common  mea- 
sure of  18  and  24 ;  the  quotient  of  the  former  being  3,  and  of  the 
latter  4,  which  will  not  both  divide  farther. 

RULE. — If  there  be  two  numbers  only,  divide  the  greater  by 
the  less  ;  then  divide  the  divisor  by  the  remainder ;  and  so  on,  divid- 
ing always  the  last  divisor  by  the  last  remainder,  till  nothing 
remains ;  then  shall  the  last  divisor  of  all  be  the  greatest  common 
measure  sought. 

When  there  are  more  than  two  numbers,  find  the  greatest  com- 
mon measure  of  two  of  them,  as  before ;  then  do  the  same  for  that 
common  measure  and  another  of  the  numbers ;  and  so  on,  through 
all  the  numbers ;  then  will  the  greatest  common  measure  last  found 
be  the  answer. 

If  it  happen  that  the  common  measure  thus  found  is  1,  then  the 
numbers  are  said  to  be  incommensurable,  or  to  have  no  common 
measure. 


BEDUCTION   OF  FRACTIONS.  17 

To  find  the  greatest  common  measure  of  1998,  918,  and  522. 
918  )  1998  (2  So  54  is  the  greatest  common  measure 

1836  of  1998  and  918. 

"162 )  918  ( 5        Hence  54 )  522  ( 9 
810  486 

108)162(1  36)54(1 

108  36 

54)108(2  18)36(2 

108  36 

So  that  18  is  the  answer  required. 

To  abbreviate  or  reduce  fractions  to  their  lowest  terms. 

RULE. — Divide  the  terms  of  the  given  fraction  by  any  number 
that  will  divide  them  without  a  remainder ;  then  divide  these  quo- 
tients again  in  the  same  manner ;  and  so  on,  till  it  appears  that 
there  is  no  number  greater  than  1  which  will  divide  them ;  then  the 
fraction  will  be  in  its  lowest  terms. 

Or,  divide  both  the  terms  of  the  fraction  by  their  greatest  com- 
mon measure,  and  the  quotients  will  be  the  terms  of  the  fraction 
required,  of  the  same  value  as  at  first. 

That  dividing  both  the  terms  of  the  fraction  by  the  same  num- 
ber, whatever  it  be,  will  give  another  fraction  equal  to  the  former, 
is  evident.  And  when  those  divisions  are  performed  as  often  as 
can  be  done,  or  when  the  common  divisor  is  the  greatest  possible, 
the  terms  of  the  resulting  fraction  must  be  the  least  possible. 

1.  Any  number  ending  with  an  even  number,  or  a  cipher,  is  divi- 
sible, or  can  be  divided  by  2. 

2.  Any  number  ending  with  5,  or  0,  is  divisible  by  5. 

3.  If  the  right-hand  place  of  any  number  be  0,  the  whole  is 
divisible  by  10 ;  if  there  be  2  ciphers,  it  is  divisible  by  100 ;  if 
3  ciphers,  by  1000 ;    and  so  on,  which  is  only  cutting  off  those 
ciphers. 

4.  If  the  two  right-hand  figures  of  any  number  be  divisible 
by  4,  the  whole  is  divisible  by  4.     And  if  the  three  right-hand 
figures  be  divisible  by  8,  the  whole  is  divisible  by  8 ;  and  so  on. 

5.  If  the  sum  of  the  digits  in  any  number  be  divisible  by  3,  or 
by  9,  the-whole  is  divisible  by  3,  or  by  9. 

6.  If  the  right-hand  digit  be  even,  .and  the  sum  of  all  the  digits 
be  divisible  by  6,  then  the  whole  will  be  divisible  by  6. 

7.  A  number  is  divisible  by  11  when  the  sum  of  the  1st,  3d, 
5th,  &c.,  or  of  all  the  odd  places,  is  equal  to  the  sum  of  the  2d, 
4th,  6th,  &c.,  or  of  all  the  even  places  of  digits. 

8.  If  a  number  cannot  be  divided  by  some  quantity  less  than 
the  square  of  the  same,  that  number  is  a  prime,  or  cannot  be 
divided  by  any  number  whatever. 

9.  All  prime  numbers,  except  2  and  5,  have  either  1,  3,  7,  or  9, 
in  the  place  of  units ;  and  all  other  numbers  are  composite,  or  can, 
be  divided. 

u2  2 


18  THE   PRACTICAL   MODEL   CALCULATOR. 

10.  When  numbers,  with  a  sign  of  addition  or  subtraction  between 
them,  are  to  be  divided  by  any  number,  then  each  of  those  num- 
bers must  be  divided  by  it.     Thus,  -       2       -=5  +  4  —  2  =  7. 

11.  But  if  the  numbers  have  the  sign  of  multiplication  between 

•I  A    vx    Q    y    q 

them,  only  one  of  them  must  be  divided.     Thus, — ^ — ^ —  = 

10  x4x3  _1Q  x4x~L     10  x2xl      20 
6x1  2x1  Ixl        =T= 

Reduce  ||f  to  its  least  terms. 

Mi  =  i¥o  =  i*=  «  =  ft  =  f,  the  answer. 
Or  thus : 

144 )  240  ( 1  Therefore  48  is  the  greatest  common  measure,  and 
144  48 )  |ff  =  f  the  answer,  the  same  as  before. 

"96)144(1 

96 

48)96(2 
96 

To  reduce  a  mixed  number  to  its  equivalent  improper  fraction. 
RULE. — Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  add  the  numerator  to  the  product ;  then  set  that  sum 
above  the  denominator  for  the  fraction  required. 
Reduce  23f  to  a  fraction. 

Or, 

23        (23  x  5)  +  2  _  117 
__5  .5         ""    5  ' 

115 
2 

Tff 
5 

To  reduce  an  improper  fraction  to  its  equivalent  whole  or  mixed 

number. 

RULE. — Divide  the  numerator  by  the  denominator,  and  the  quo- 
tient will  be  the  whole  or  mixed  number  sought.  * 

Reduce  ^  to  its  equivalent  number. 

Here  V  or  12  -^  3  =  4. 
Reduce  Vs  to  its  equivalent  number. 

Here  V  or  15  -^  7  =  2f 
Reduce  %9  to  its  equivalent  number. 

Thus,  17)  749  (44£ 
68^ 

69"  So  that  %9 

68 
1 


REDUCTION   OF  FRACTIONS.  19 

To  reduce  a  ivhole  number  to  an  equivalent  fraction,  having  a 

given  denominator. 

RULE. — Multiply  the  whole  number  by  the  given  denominator, 
then  set  the  product  over  the  said  denominator,  and  it  will  form 
the  fraction  required. 

Reduce  9  to  a  fraction  whose  denominator  shall  be  7. 

Here  9  x  7  =  63,  then  6T3  is  the  answer. 
For  y  =  63  H-  7  =  9,  the  proof. 

To  reduce  a  compound  fraction  to  an  equivalent  simple  one. 

RULE. — Multiply  all  the  numerators  together  for  a  numerator, 
and  all  the  denominators  together  for  the  denominator,  and  they 
will  form  the  simple  fraction  sought. 

.When  part  of  the  compound  fraction  is  a  whole  or  mixed  number, 
it  must  first  be  reduced  to  a  fraction  by  one  of  the  former  cases. 

And,  when  it  can  be  done,  any  two  terms  of  the  fraction  may  be 
divided  by  the  same  number,  and  the  quotients  used  instead  of 
them.  Or,  when  there  are  terms  that  are  common,  they  may  be 
omitted. 

Reduce  J  of  f  of  }  to  a  simple  fraction. 

1x2x3       6        1 

Here  2x3x4==24==4' 
1x2x3       1  , 

Or,  o  x  Q  x  4  =  I>  °y  omitting  the  twos  and  threes. 

Reduce  f  of  f  of  \\  to  a  simple  fraction. 

2  x  3  x  10        60        12        4 

Here  3  x  5  x  11  =  165  =  33  =  if 
^     2  x  3  x  10        4 

Or,  3  x  5  x  1 1  =  TT>  tne  same  as  before. 

To  reduce  fractions  of  different  denominators  to -equivalent  frac- 
tions, having  a  common  denominator. 

RULE. — Multiply  each  numerator  into  all  the  denominators  ex- 
cept its  own  for  the  new  numerators ;  and  multiply  all  the  denomi- 
nators together  for  a  common  denominator. 

It  is  evident,  that  in  this  and  several  other  operations,  when  any 
of  the  proposed  quantities  are  integers,  or  mixed  numbers,  or  com- 
pound fractions,  they  must  be  reduced,  by  their  proper  rules,  to 
the  form  of  simple  fractions. 

Reduce  J,  f ,  and  }  to  a  common  denominator. 

1  X  3  X  4  =  12  the  new  numerator  for  J. 

2x2x4  =  16 for  f. 

3x2x3  =  18 for  }. 

2  X  3  X  4  =  24  the  common  denominator. 
Therefore,  the  equivalent  fractions  are  Jf,  |f,  and  Jf. 

Or,  the  whole  operation  of  multiplying  may  be  very  well  per- 
formed mentally,  and  only  set  down  the  results  and  given  fractions 
thus :  J,  |,  4  ==  Jf,  |f,  if  =  A,  A,  &  by  abbreviation. 


20  THE  PRACTICAL  MODEL  CALCULATOR. 

When  the  denominators  of  two  given  fractions  have  a  common 
measure,  let  them  be  divided  by  it ;  then  multiply  the  terms  of 
each  given  fraction  by  the  quotient  arising  from  the  other's  deno- 
minator. 

When  the  less  denominator  of  two  fractions  exactly  divides  the 
greater,  multiply  the  terms  of  that  which  hath  the  less  denominator 
by  the  quotient. 

When  more  than  two  fractions  are  proposed,  it  is  sometimes  con- 
venient first  to  reduce  two  of  them  to  a  common  denominator, 
then  these  and  a  third ;  and  so  on,  till  they  be  all  reduced  to  then- 
least  common  denominator. 

To  find  the  value  of  a  fraction  in  parts  of  the  integer. 

EULE. — Multiply  the  integer  by  the  numerator,  and  divide  the 
product  by  the  denominator,  by  Compound  Multiplication  and 
Division,  if  the  integer  be  a  compound  quantity. 

Or,  if  it  be^a  single  integer,  multiply  the  numerator  by  the  parts 
in  the  next  inferior  denomination,  and  divide  the  product  by  the 
denominator.  Then,  if  any  thing  remains,  multiply  it  by  the  parts 
in  the  next  inferior  denomination,  and  divide  by  the  denominator 
as  before ;  and  so  on,  as  far  as  necessary ;  so  shall  the  quotients, 
placed  in  order,  be  the  value  of  the  fraction  required. 

What  is  the  value  of  f  of  a  pound  troy  ?  7  oz.  4  dwts. 

What  is  the  value  of  -^  of  a  cwt.?  1  qr.  7  Ib. 

What  is  the  value  of  f  of  an  acre  ?  2  ro.  20  po. 

What  is  the  value  of  ^  of  a  day?  7  hrs.  12  min. 

To  reduce  a  fraction  from  one  denomination  to  another. 
RULE. — Consider  how  many  of  the  less   denomination   make 
one  of  the  greater;   then  multiply  the  numerator  by  that  num- 
ber, if  the  reduction  be  to  a  less  name,  or  the  denominator,  if  to 
a  greater. 

Reduce  f  of  a  cwt.  to  the  fraction  of  a  pound. 

I  x  |  x  y  =  y. 

ADDITION  OF  FRACTIONS. 
To  add  fractions  together  that  have  a  common  denominator. 

RULE. — Add  all  the  numerators  together,  and  place  the  sum 
over  the  common  denominator,  and  that  will  be  the  &um  of  the 
fractions  required. 

If  the  fractions  proposed  have  not  a  common  denominator,  they 
must  be  reduced  to  one.  Also,  compound  fractions  must  be  reduced 
to  simple  ones,  and  mixed  numbers  to  improper  fractions;  also, 
fractions  of  different  denominations  to  those  of  the  same  denomi- 
nation. 

To  add  f  and  |  together.  Here  f  -f  |  =  {  =  If- 

To  add  f  and  f  together.  f  +  f  =  ft  +  If  =  «  =  Itf. 

To  add  |  and  7£  and  \  of  j  together. 

i  +  7i  +  i  of  f  =  |  +  V  +  J  =  I .  +  6i°  +  I  =  V  =  8|. 


RULE   OF  THREE   IN   FRACTIONS.  21 

SUBTRACTION   OF   FRACTIONS. 

RULE.  —  Prepare  the  fractions  the  same  as  for  Addition  ;  then  sub- 
tract the  one  numerator  from  the  other,  and  set  the  remainder  over 
the  common  denominator,  for  the  difference  of  the  fractions  sought. 
To  find  the  difference  between  f  and  J. 

Here  f  -  J  =  J  =  |. 
To  find  the  difference  between  f  and  f 

!  -  f  =  If  -  \  1  =  A* 

MULTIPLICATION  OF   FRACTIONS. 

MULTIPLICATION  of  any  thing  by  a  fraction  implies  the  taking 
some  part  or  parts  of  the  thing  ;  it  may  therefore  be  truly  expressed 
by  a  compound  fraction  ;  which  is  resolved  by  multiplying  together 
the  numerators  and  the  denominators. 

RULE.  —  Reduce  mixed  numbers,  if  there  be  any,  to  equivalent 
fractions  ;  then  multiply  all  the  numerators  together  for  a  nume- 
rator, and  all  the  denominators  together  for  a  denominator,  which 
will  give  the  product  required. 

Required  the  product  of  f  and  f. 

Here  f  X  f-  =  &  =  J. 
Or,  f  X  |  =  \  X  I  =  i. 

Required  the  continued  product  of  f,  3J,  5,  and  f  of  f. 

2      13       5       3      3       13  x  3       39 
XxxX--          =  4f. 


DIVISION  OF  FRACTIONS. 

RULE.  —  Prepare  the  fractions  as  before  in  Multiplication  ;  then 
divide  the  numerator  by  the  numerator,  and  the  denominator  by 
the  denominator,  if  they  will  exactly  divide  ;  but  if  not,  then  invert 
the  terms  of  the  divisor,  and  multiply  the  dividend  by  it,  as  in 
Multiplication. 

Divide  ^  by  f. 

Here  f  ~  f  =  |  =  If,  by  the  first  method. 
Divide  f  by  &. 

Here  f  -J-  &  =  f  x  tf  =  f  x  f  =  f  =  4J,  by  the  latter. 

RULE   OF  THREE   IN  FRACTIONS. 

RULE.  —  Make  the  necessary  preparations  as  before  directed  ; 
then  multiply  continually  together  the  second  and  third  terms,  and 
the  first  with  its  terms  inverted  as  in  Division,  for  the  answer. 
This  is  only  multiplying  the  second  and  third  terms  together,  and 
dividing  the  product  by  the  first,  as  in  the  Rule  of  Three  in  whole 
numbers. 

If  f  of  a  yard  of  velvet  cost  f  of  a  dollar,  what  will  ^  of  a 
yard  cost  ? 

32       582        5 
Here  g:5:;Jg:3x5XJg==iofa  dollar. 


22  THE   PRACTICAL   MODEL   CALCULATOR. 

DECIMAL  FRACTIONS. 

A  DECIMAL  FRACTION  is  that  which  has  for  its  denominator  a 
unit  (1)  with  as  many  ciphers  annexed  as  the  numerator  has  places ; 
and  it  is  usually  expressed  by  setting  down  the  numerator  only, 
with  a  point  before  it  on  the  left  hand.  Thus,  ^  is  -5,  and  ^  is 
•25,  and  ^  ™  'W5,  and  ^sW  is  -00124 ;  where  ciphers  are  pre- 
fixed to  make  up  as  many  places  as  are  in  the  numerator,  when 
there  is  a  deficiency  of  figures. 

A  mixed  number  is  made  up  of  a  whole  number  with  some  deci- 
mal fraction,  the  one  being  separated  from  the  other  by  a  point. 
Thus,  3*25  is  the  same  as  3^,  or  fgj. 

Ciphers  on  the  right  hand  of  decimals  make  no  alteration  in 
their  value;  for  *5,  or  *50,  or  '500,  are  decimals  having  all  the 
same  value,  being  each  =  ^  or  J.  But  if  they  are  placed  on  the 
left  hand,  they  decrease  the  value  in  a  tenfold  proportion.  Thus, 
•5  is  ^  or  5  tenths,  but  '05  is  only  ^  or  5  hundreths,  and  '005 
is  but  yj&fl  or  5  thousandths. 

The  first  place  of  decimals,  counted  from  the  left  hand  towards 
the  right,  is  called  the  place  of  primes,  or  lOths ;  the  second  is  the 
place  of  seconds,  or  lOOths;  the  third  is  the  place  of  thirds,  or 
lOOOths ;  and  so  on.  For,  in  decimals,  as  well  as  in  whole  num- 
bers, the  values  of  the  places  increase  towards  the  left  hand,  and 
decrease  towards  the  right,  both  in  the  same  tenfold  proportion ; 
as  in  the  following  Scale  or  Table  of  Notation : 


I  f     1    3     1     J 

§        |  r        &       I      1        1       1 

a     §     »  jT   *    a     I    5     * 

^^s-g-s  s    £    -«     §    ^    6 

.1       I       :S         §       I          .        »•  4       j  Sll 

1  I  i  I  I  I  i    1  1  i  i  I  i 

3333333         333333 

ADDITION  OF  DECIMALS. 

RULE. — Set  the  numbers  under  each  other  according  to  the  value 
of  their  places,  like  as  in  whole  numbers ;  in  which  state  the  deci- 
mal separating  points  will  stand  all  exactly  under  each  other. 
Then,  beginning  at  the  right  hand,  add  up  all  the  columns  of 
number  as  in  integers,  and  point  off  as  many  places  for  decimals  as 
are  in  the  greatest  number  of  decimal  places  in  any  of  the  lines  that 
are  added ;  or,  place  the  point  directly  below  all  the  other  points. 
To  add  together  29-0146,  and  3146-5,  29-0146 

and  2109,  and  62417,  and  14-16.    '         3146-5 

2109- 

•62417 
14-16 

5299-29877,  the  sum. 


MULTIPLICATION   OF   DEC! 

The  sum  of  376-25  +  86-125  +,  637-4725 
358-865  =  1506.2325. 

The  sum  of  3-5  +  47-25  -f-  2.0073  +  927-01  +  1-5  =  981.2673. 

The  sum  of  276  +  54-321  +  112  +  0.65  +  12-5  +  '0463  = 
455-5173. 

SUBTRACTION  OF  DECIMALS. 

RULE. — Place  the  numbers  under  each  other  according  to  the 
value  of  their  places,  as  in  the  last  rule.  Then,  beginning  at  the 
right  hand,  subtract  as  in  whole  numbers,  and  point  off  the  deci- 
mals as  in  Addition. 


To  find  the  difference  between 
91.73  and  2.138. 


91-73 

2-138 


89-592  the  difference. 
The  difference  between  1-9185  and  2-73  =  0-8115. 
The  difference  between  214-81  and  4-90142  =  209-90858. 
The  difference  between  2714  and  -916  =  2713-084. 

MULTIPLICATION  OF  DECIMALS. 


RULE. — Place  the  factors,  and 
multiply  them  together  the  same 
as  if  they  were  whole  numbers. 
Then  point  off  in  the  product  just 
as  many  places  of  decimals  as 
there  are  decimals  in  both  the  fac- 
tors. But  if  there  be  not  so  many 
figures  in  the  product,  then  supply 
the  defect  by  prefixing  ciphers. 


Multiply  *321096 
by      -2465 
1605480 
1926576 
1284384 
642192 


•0791501640  the  product 


Multiply  79-347  by  23-15,  and  we  have  1836-88305. 
Multiply  -63478  by  -8204,  and  we  have  -520773512. 
Multiply  -385746  by  -00464,  and  we  have  -00178986144. 


CONTRACTION   I. 


To  multiply  decimals  by  1  with  any  number  of  ciphers,  as  10,  or 

100,  or  1000,  fe. 

This  is  done  by  only  removing  the  decimal  point  so  many  places 
farther  to  the  right  hand  as  there  are  ciphers  in  the  multiplier ; 
and  subjoining  ciphers  if  need  be. 

The  product  of     51-3  and    1000  is  51300. 
The  product  of  2-714  and      100  is  271-4. 
The  product  of     -916  and    1000  is  916. 
The  product  of  21-31  and  10000  is  213100. 


CONTRACTION  II. 


To  contract  the  operation,  so  as  to  retain  only  as  many  decimals  in 
the  product  as  may  be  thought  necessary,  when  the  product  would 
naturally  contain  several  more  places. 

Set  the  units'  place  of  the  multiplier  under  that  figure  of  the 
multiplicand  whose  place  is  the  same  as  is  to  be  retained  for  the 


24  THE  PRACTICAL  MODEL  CALCULATOR. 

last  in  the  product ;  and  dispose  of  the  rest  of  the  figures  in  the 
inverted  or  contrary  order  to  what  they  are  usually  placed  in. 
Then,  in  multiplying,  reject  all  the  figures  that  are  more  to  the 
right  than  each  multiplying  figure ;  and  set  down  the  products,  so 
that  their  right  hand  figures  may  fall  in  a  column  straight  below 
each  other ;  but  observing  to  increase  the  first  figure  of  every  line 
with  what  would  arise  from  the  figures  omitted,  in  this  manner, 
namely,  1  from  5  to  14,  2  from  15  to  24,  3  from  25  to  34,  &c. ; 
and  the  sum  of  all  the  lines  will  be  the  product  as  required,  com- 
monly to  the  nearest  unit  in  the  last  figure. 

To  multiply  27-14986  by  92-41035,  so  as  to  retain  only  four 
places  of  decimals  in  the  product. 

Contracted  way.  Common  way. 

27-14986    '  27-14986 

53014-29  92-41035 


24434874  13 


542997  81 44958 

108599  2714986 

2715  10859944 

81  542997 

14  24434874 


574930 


2508-9280  2508-9280  650510 

DIVISION  OF  DECIMALS. 

RULE. — Divide  as  in  whole  numbers ;  and  point  off  in  the  quo- 
tient as  many  places  for  decimals,  as  the  decimal  places  in  the 
dividend  exceed  those  in  the  divisor. 

When  the  places  of  the  quotient  are  not  so  many  as  the  rule  re- 
quires, let  the  defect  be  supplied  by  prefixing  ciphers. 

When  there  happens  to  be  a  remainder  after  the  division ;  or 
when  the  decimal  places  in  the  divisor  are  more  than  those  in  the 
dividend ;  then  ciphers  may  be  annexed  to  the  dividend,  and  the 
quotient  carried  on  as  far  as  required. 

179)  -48624097  (-00271643    |    -2685)27-00000(100-55865 

1282  15000 

294  15750 

1150  23250 

769  17700 

537  15900 

OOP  24750 

Divide  234-70525  by  64-25.  3-653. 

Divide  14  by  -7854.  17-825. 

Divide  2175-68  by  100.  21-7568. 

Divide  -8727587  by  -162.  5-38739. 

CONTRACTION   I. 

When  the  divisor  is  an  integer,  with  any  number  of  ciphers  an- 
nexed ;  cut  off  those  ciphers,  and  remove  the  decimal  point  in  the 


REDUCTION  OF  DECIMALS.  25 

dividend  as  many  places  farther  to  the  left  as  there  are  ciphers  cut 
off,  prefixing  ciphers  if  need  be  ;  then  proceed  as  before. 
Divide  45-5  by  2100. 

21-00) -455  (-0216,  &c. 
35 
140 
14 

CONTRACTION   II. 

Hence,  if  the  divisor  be  1  with  ciphers,  as  10,  or  100,  or  1000, 
&c. ;  then  the  quotient  will  be  found  by  merely  moving  the  decimal 
point  in  the  dividend  so  many  places  farther  to  the  left  as  the  di- 
visor has  ciphers  ;  prefixing  ciphers  if  need  be. 

So,  217-3    -r-  100  =  2-173,         and  419  -r-      10  =  41-9. 

And    5-16  -T-  100  =  -0516,         and  -21  -v-  1000  =  -00021. 

CONTRACTION   III. 

"When  there  are  many  figures  in  the  divisor ;  or  only  a  certain 
number  of  decimals  are  necessary  to  be  retained  in  the  quotient, 
then  take  only  as  many  figures  of  the  divisor  as  will  be  equal  to 
the  number  of  figures,  both  integers  and  decimals,  to  be  in  the  quo- 
tient, and  find  how  many  times  they  may  be  contained  in  the  first 
figures  of  the  dividend,  as  usual. 

Let  each  remainder  be  a  new  dividend ;  and  for  every  such  divi- 
dend, leave  out  one  figure  more  on  the  right  hand  side  of  the  di- 
visor ;  remembering  to  carry  for  the  increase  of  the  figures  cut  off, 
as  in  the  2d  contraction  in  Multiplication. 

When  there  are  not  so  many  figures  in  the  divisor  as  are  required 
to  be  in  the  quotient,  begin  the  operation  with  all  the  figures,  and 
continue  it  as  usual  till  the  number  of  figures  in  the  divisor  be  equal 
to  those  remaining  to  be  found  in  the  quotient,  after  which  begin 
the  contraction. 

Divide  2508-92806  by  92-41035,  so  as  to  have  only  four  deci- 
mals in  the  quotient,  in  which  case  the  quotient  will  contain  six 
figures. 

Contracted.  Common  way. 


92-4103,5)  2508-928,06  (27-1498 

660721 

13849 

4608 

912 

80 

6 


92-4103,5)  2508-928,06  (27-1498 
66072106 
13848610 
46075750 
91116100 
79467850 
5539570 


REDUCTION  OF  DECIMALS. 

To  reduce  a  common  fraction  to  its  equivalent  decimal. 
RULE. — Divide  the  numerator  by  the  denominator  as  in  Division 
of  Decimals,  annexing  ciphers  to  the  numerator  as  far  as  necessary; 
so  shall  the  quotient  be  the  decimal  required. 


26  THE  PRACTICAL  MODEL  CALCULATOR. 

Reduce  ^  to  a  decimal. 

24  =  4  x  6.         Then  4  )  7- 

6)1-750000 

•291666,  &c. 

f  reduced  to  a  decimal,  is  -375. 

•^  reduced  to  a  decimal,  is  -04. 

if-2  reduced  to  a  decimal,  is  -015625. 

$£z  reduced  to  a  decimal,  is  -071577,  &c. 

CASE   II. 

To  find  the  value  of  a  decimal  in  terms  of  the  inferior  denominations. 

RULE. — Multiply  the  decimal  by  the  number  of  parts  in  the  next 
lower  denomination ;  and  cut  off  as  many  places  for  a  remainder, 
to  the  right  hand,  as  there  are  places  in  the  given  decimal. 

Multiply  that  remainder  by  the  parts  in  the  next  lower  denomi- 
nation again,  cutting  off  for  another  remainder  as  before. 

Proceed  in  the  same  manner  through  all  the  parts  of  the  integer ; 
then  the  several  denominations,  separated  on  the  left  hand,  will 
make  up  the  value  required. 

What  is  the  value  of  -0125  Ib.  troy :—  3  dwts. 

What  is  the  value  of  -4694  Ib.  troy :—  5  oz.  12  dwt.  15-744  gr. 

What  is  the  value  of  -625  cwt. :—  2  qr.  14  Ib. 

What  is  the  value  of  -009943  miles :—     17  yd.  1  ft.  5-98848  in. 

What  is  the  value  of  -6875  yd. :—  2  qr.  3  nls. 

What  is  the  value  of  -3375  ac. : —  1  rd.  14  poles. 

What  is  the  value  of  -2083  hhd.  of  wine :—  13-1229  gal. 

CASE   III. 

To  reduce  integers  or  decimals  to  equivalent  decimals  of  higher 
denominations. 

RULE. — Divide  by  the  number  of  parts  in  the  next  higher  de- 
nomination ;  continuing  the  operation  to  as  many  higher  denomi- 
nations as  may  be  necessary,  the  same  as  in  Reduction  Ascending 
of  whole  numbers. 

Reduce  1  dwt.  to  the  decimal  of  a  pound  troy. 


20 
12 


1  dwt. 
0-05  oz. 
0-004166,  &c.  Ib. 


Reduce  7  dr.  to  the  decimal  of  a  pound  avoird.:—  -02734375  Ib. 

Reduce  2-15  Ib.  to  the  decimal  of  a  cwt. :—  -019196  cwt. 

Reduce  24  yards  to  the  decimal  of  a  mile: —  -013636,  &c.  miles.. 

Reduce  -056  poles  to  the  decimal  of  an  acre  : —  -00035  ac. 

Reduce  1-2  pints  of  wine  to  the  decimal  of  a  hhd. : —  -00238  hhd. 

Reduce  14  minutes  to  the  decimal  of  a  day : —   -009722,  &c.  da. 

Reduce  -21  pints  to  the  decimal  of  a  peck: —  -013125  pec. 

When  there  are  several  numbers,  to  be  reduced  all  to  the  decimal  of 

the  highest. 

Set  the  given  numbers  directly  under  each  other,  for  dividends, 
proceeding  orderly  from  the  lowest  denomination  to  the  highest. 


DUODECIMALS.  27 

Opposite  to  each  dividend,  on  the  left  hand,  set  such  a  number 
for  a  divisor  as  will  bring  it  to  the  next  higher  name ;  drawing  a 
perpendicular  line  between  all  the  divisors  and  dividends. 

Begin  at  the  uppermost,  and  perform  all  the  divisions ;  only  ob- 
serving to  set  the  quotient  of  each  division,  as  decimal  parts,  on 
the. right  hand  of  the  dividend  next  below  it ;  so  shall  the  last  quo- 
tient be  the  decimal  required. 

Reduce  5  oz.  12  dwts.  16  gr.  to  Ibs. :—  -46944,  &c.  Ib. 

RULE  OF  THREE  IN  DECIMALS. 

RULE. — Prepare  the  terms  by  reducing  the  vulgar  fractions  to 
decimals,  any  compound  numbers  either  to  decimals  of  the  higher 
denominations,  or  to  integers  of  the  lower,  also  the  first  and  third 
terms  to  the  same  name :  then  multiply  and  divide  as  in  whole 
numbers. 

Any  of  the  convenient  examples  in  the  Rule  of  Three  or  Rule  of 
Five  in  Integers,  or  Common  Fractions,  may  be  taken  as  proper 
examples  to  the  same  rules  in  Decimals. — The  following  example, 
which  is  the  first  in  Common  Fractions,  is  wrought  here  to  show  the 
method. 

If  f  of  a  yard  of  velvet  cost  f  of  a  dollar,  what  will  ^  yd.  cost  ? 

yd.      $          yd.         $ 
I  =  -375       -375  :  -4  : :  -3125  :  -333,  &c. 
-4 

f  =  -4  -375 )  ^12500  ( -333333,  33J  cts. 

1250 
125 
A  =  -3125. 

DUODECIMALS. 

DUODECIMALS,  or  CROSS  MULTIPLICATION,  is  a  rule  made  use  of 
by  workmen  and  artificers,  in  computing  the  contents  of  their  works. 

Dimensions  are  usually  taken  in  feet,  inches,  and  quarters  ;  any 
parts  smaller  than  these  being  neglected  as  of  no  consequence. 
And  the  same  in  multiplying  them  together,  or  casting  up  the  con- 
tents. 

RULE. — Set  down  the  two  dimensions,  to  be  multiplied  together, 
one  under  the  other,  so  that  feet  stand  under  feet,  inches  under 
inches,  &c. 

Multiply  each  term  in  the  multiplicand,  beginning  at  the  lowest, 
by  the  feet  in  the  multiplier,  and  set  the  result  of  each  straight  un- 
der its  corresponding  term,  observing  to  carry  1  for  every  12,  from 
the  inches  to  the  feet. 

In  like  manner,  multiply  all  the  multiplicand  by  the  inches  and 
parts  of  the  multiplier,  and  set  the  result  of  each  term  one  place 
removed  to  the  right  hand  of  those  in  the  multiplicand ;  omitting, 
however,  what  is  below  parts  of  inches,  only  carrying  to  these  the 
proper  number  of  units  from  the  lowest  denomination. 


28 


THE  PRACTICAL  MODEL  CALCULATOR. 


Or,  instead  of  multiplying  by  the  inches,  take  such  parts  of  the 
multiplicand  as  these  are  of  a  foot.' 

Then  add  the  two  lines  together,  after  the  manner  of  Compound  Ad- 
dition, carrying  1  to  the  feet  for  12  inches,  when  these  come  to  so  many. 

Multiply  4  f.  7  inc.  Multiply  14  f.  9  inc. 

by  6      4  by    4      6 


27 
1 


6 


29      OJ 


INVOLUTION, 

INVOLUTION  is  the  raising  of  Powers  from  any  given  number,  as 
a  root. 

A  Power  is  a  quantity  produced  by  multiplying  any  given  num- 
ber, cabled  the  Root,  a  certain  number  of  times  continually  by 
itself.     Thus,          2  =    2  is  the  root,  or  first  power  of  2. 
2x2=    4  is  the  2d  power,  or  square  of  2. 
2x2x2=    8  is  the  3d  power,  or  cube  of  2. 
2  X  2  x  2  X  2  =  16  is  the  4th  power  of  2,  &c. 

And  in  this  manner  may  be  calculated  the  following  Table  of  the 
first  nine  powers  of  the  first  nine  numbers. 

TABLE   OF   THE   FIRST   NINE   POWERS   OF   NUMBERS. 


1st 

-2d. 

3d. 

4th. 

5th. 

6th. 

7th. 

8th. 

9th. 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

128 

256 

512 

3 

9 

27 

81 

243 

729 

2187 

6561 

19683 

4 

16 

64 

256 

1024 

4096 

16384 

65536 

262144 

5 

25 

125 

625 

3125 

15625 

78125 

390625 

1953125 

6 

36 

216 

1296 

7776 

46656 

279936 

1679616 

10077696 

7 

T 

~9~ 

49 

343 

2401 

16807 

117649 

823543 

5764801 

40353607 

64 
81 

512 

4096 

32768 

262144 

2097152 

16777216 

134217728 

729 

6561 

59049 

531441 

4782969 

43046721 

387420489 

The  Index  or  Exponent  of  a  Power  is  the  number  denoting  the 
height  or  degree  of  that  power ;  and  it  is  1  more  than  the  number 
of  multiplications  used  in  producing  the  same.  So  1  is  the  index 
or  exponent  of  the  1st  power  or  root,  2  of  the  2d  power  or  square, 
3  of  the  3d  power  or  cube,  4  of  the  4th  power,  and  so  on. 

Powers,  that  are  to  be  raised,  are  usually  denoted  by  placing  the 
index  above  the  root  or  first  power. 

So  22  =    4,     is  the  2d  power  of  2. 

23  =    8,     is  the  3d  power  of  2. 

24  =  16,     is  the  4th  power  of  2. 

5404,     is  the  4th  power  of  540  =  85030560000. 


EVOLUTION.  29 

When  two  or  more  powers  are  multiplied  together,  tlieir  product 
will  be  that  power  whose  index  is  the  sum  of  the  exponents  of  the 
factors  or  powers  multiplied.  Or,  the  multiplication  of  the  powers 
answers  to  the  addition  of  the  indices.  Thus,  in  the  following 
powers  of  2. 

1st.  2d.  3d.  4th.  5th.  6th.  7th.     8th.    9th.    10th. 

2      4     8     16     32     64     128     256     512     1024 

or,  21     22    23      24      25     26       27       28       29         210 

Here,  4x4=  16,  and  2  -f  2  =  4  its  index  ; 
and  8x16=  128,  and  3  +  4  =  7  its  index ; 
also  16  X  64  =  1024,  and  4  -f  6  =  10  its  index. 

The  2d  power  of  45  is  2025. 

The  square  of  4-16  is  17*3056. 

The  3d  power  of  3-5  is  42-875. 

The  5th  power  of  *029  is  -000000020511149. 

The  square  of  §  is  f. 

The  3d  power  of  f  is  f||. 

The  4th  power  of  f  is  ^. 

EVOLUTION. 

EVOLUTION,  or  the  reverse  of  Involution,  is  the  extracting  or 
finding  the  roots  of  any  given  powers. 

The  root  of  any  number,  or  power,  is  such  a  number  as,  being 
multiplied  into  itself  a  certain  number  of  times,  will  produce  that 
power.  Thus,  2  is  the  square  root  or  2d  root  of  4,  because  22  = 

2  x  2  =  4 ;  and  3  is  the  cube  root  or  3d  root  of  27,  because  33  = 

3  x  3  x  3  =  27. 

Any  power  of  a  given  number  or  root  may  be  found  exactly, 
namely,  by  multiplying  the  number  continually  into  itself.  But 
there  are  many  numbers  of  which  a  proposed  root  can  never  be 
exactly  found.  Yet,  by  means  of  decimals  we  may  approximate 
or  approach  towards  the  root  to  any  degree  of  exactness. 

These  roots,  which  only  approximate,  are  called  Surd  roots  ;  but 
those  which  can  be  found  quite  exact,  are  called  Rational  roots. 
Thus,  the  square  root  of  3  is  a  surd  root ;  but  the  square  root  of 

4  is  a  rational  root,  being  equal  to  2 :  also,  the  cube  root  of  8  is 
rational,  being  equal  to  2 ;    but  the  cube  root  of  9  is  surd,  or 
irrational. 

Roots  are  sometimes  denoted  by  writing  the  character  \/  before 
the  power,  with  the  index  of  the  root  against  it.  Thus,  the  third 
root  of  20  is  expressed  by  ^20 ;  and  the  square  root  or  2d  root 
of  it  is  N/20,  the  index  2  being  always  omitted  when  the  square 
root  is  designed. 

When  the  power  is  expressed  by  several  numbers,  with  the  sign 
+  or  —  between  them,  a  line  is  drawn  from  the  top  of  the  sign  over 
all  the  parts  of  it ;  thus,  the  third  root  of  45  —  12  is  ^45  —  12", 
or  thus,  -\K(45  —  12),  enclosing  the  numbers  in  parentheses, 
c  2 


30  THE  PRACTICAL  MODEL  CALCULATOR. 

But  all  roots  are  now  often  designed  like  powers,  with  fractional 
indices  :  thus,  the  square  root  of  8  is  8*,  the  cube  root  of  25  is  25  , 
and  the  4th  root  of  45  -  18  is  45  -  18ll,  or,  (45  -  18)1 

TO   EXTRACT   THE   SQUARE  ROOT. 

RULE. — Divide  the  given  number  into  periods  of  two  figures 
each,  by  setting  a  point  over  the  place  of  units,  another  over  the 
place  of  hundreds,  and  so  on,  over  every  second  figure,  both  to  the 
left  hand  in  integers,  and  to  the  right  in  decimals. 

Find  the  greatest  square  in  the  first  period  on  the  left  hand,  and 
set  its  root  on  the  right  hand  of  the  given  number,  after  the  man- 
ner of  a  quotient  figure  in  Division. 

Subtract  the  square  thus  found  from  the  said  period,  and  to  the 
remainder  annex  the  two  figures  of  the  next  following  period  for  a 
dividend. 

Double  the  root  above  mentioned  for  a  divisor,  and  find  how 
often  it  is  contained  in  the  said  dividend,  exclusive  of  its  right-hand 
figure  ;  and  set  that  quotient  figure  both  in  the  quotient  and  divisor. 

Multiply  the  whole  augmented  divisor  by  this  last  quotient  figure, 
and  subtract  the  product  from  the  said  dividend,  bringing  down  to 
the  next  period  of  the  given  number,  for  a  new  dividend. 

Repeat  the  same  process  over  again,  namely,  find  another  new 
divisor,  by  doubling  all  the  figures  now  found  in  the  root ;  from 
which,  and  the  last  dividend,  find  the  next  figure  of  the  root  as 
before,  and  so  on  through  all  the  periods  to  the  last. 

The  best  way  of  doubling  the  root  to  form  the  new  divisor  is  by 
adding  the  last  figure  always  to  the  last  divisor,  as  appears  in  the 
following  examples.  Also,  after  the  figures  belonging  to  the  given 
number  are  all  exhausted,  the  operation  may  be  continued  into 
decimals  at  pleasure,  by  adding  any  number  of  periods  of  ciphers, 
two  in  each  period. 

To  find  the  square  root  of  29506624. 

29506624  (5432  the  root. 

25 


104 
4 


450 
416 


1083 
3 


3466 
3249 


10862 

2 


21724 
21724 


When  the  root  is  to  be  extracted  to  many  places  of  figures,  the  work 

'may  be  considerably  shortened,  thus: 

Having  proceeded  in  the  extraction  after  the  common  method  till 
there  be  found  half  the  required  number  of  figures  in  the  root,  or 
one  figure  more ;  then,  for  the  rest,  divide  the  last  remainder  by 


TO  EXTRACT  THE  SQUARE  ROOT.  31 

its  corresponding  divisor,  after  the  manner  of  the  third  contraction 
in  Division  of  Decimals ;  thus, 

To  find  the  root  of  2  to  nine  places  of  figures. 
2(1-4142 

24  I  100 
4       96 


281 


400 

281 


2824 
4 


11900 
11296 


28282 


60400 
56564 


28284)     3836(1356 

1008 

160 

19 

2 

141421356  the  root  required. 
The  square  root  of  -000729  is  -02T. 
The  square  root  of  3  is  1-732050. 
The  square  root  of  5  is  2-236068. 
The  square  root  of  6  is  2-449489. 

EULES   FOR  THE   SQUARE  ROOTS   OP   COMMON  FRACTIONS  AND   MIXED 

NUMBERS. 

First,  prepare  all  common  fractions  by  reducing  them  to  their 
least  terms,  both  for  this  and  all  other  roots.     Then, 

1.  Take  the  root  of  the  numerator  and  of  the  denominator  for 
the  respective  terms  of  the  root  required.     And  this  is  the  best 
way  if  the  denominator  be  a  complete  power ;  but  if  it  be  not,  then, 

2.  Multiply  the  numerator  and  denominator  together ;  take  the 
root  of  the  product :  this  root  being  made  the  numerator  to  the 
denominator  of  the  given  fraction,  or  made  the  denominator  to  the 
numerator  of  it,  will  form  the  fractional  root  required. 

a       \/a       \/ab          a 

That  is,  V/T  =  —TT  =  —j—  =  —r~l- 
b        \/b  b          \/ab 

And  this  rule  will  serve  whether  the  root  be  finite  or  infinite. 

3.  Or  reduce  the  common  fraction  to  a  decimal,  and  extract  its  root. 

4.  Mixed  numbers  may  be  either  reduced  to  improper  fractions, 
and  extracted  by  the  first  or  second  rule ;  or  the  common  fraction 
may  be  reduced  to  a  decimal,  then  joined  to  the  integer,  and  the 
root  of  the  whole  extracted. 

The  root  of  ff  is  |. 

The  root  of  £.  is  f . 

The  root  of  T92  is  0-866025. 

The  root  of  ^  is  0-645497. 

The  root  of  17!  is  4-168333. 


32 


THE   PRACTICAL   MODEL   CALCULATOR. 


By  means  of  the  square  root,  also,  may  readily  be  found  the  4th 
root,  or  the  8th  root,  or  the  16th  root,  &c. ;  that  is,  the  root  of  any 
power  whose  index  is  some  power  of  the  number  2 ;  namely,  by 
extracting  so  often  the  square  root  as  is  denoted  by  that  power 
of  2 ;  that  is,  two  extractions  for  the  4th  root,  three  for  the  8th 
root,  and  so  on. 

So,  to  find  the  4th  root  of  the  number  21035-8,  extract  the 
square  root  twice  as  follows : 


21035-8000     -  ( 
1 

145-037237  (  12-0431407,  the  4th  root. 

24 
4 

110 

96 

22 

2 

45 
44 

285 
5 

1435 
1425 

2404 
4 

10372 
9616 

29003 
6 

108000 
87009 

24083 
6 

7237 

75637 
72249 

20991  ( 
687 
107 
20 

3388(1407 
980 
17 

TO  EXTRACT   THE   CUBE   ROOT. 

1.  DIVIDE  the  page  into  three  columns  (i),  (n),  (in),  in  order, 
from  left  to  right,  so  that  the  breadth  of  the  columns  may  increase 
in  the  same  order.     In  column  (in)  write  the  given  number,  and 
divide  it  into  periods  of  three  figures  each,  by  putting  a  point  over 
the  place  of  units,  and  also  over  every  third  figure,  from  thence  to 
the  left  in  whole  numbers,  and  to  the  right  in  decimals. 

2.  Find  the  nearest  less  cube  number  to  the  first  or  left-hand 
period ;  set  its  root  in  column  (in),  separating  it  from  the  right 
of  the  given  number  by  a  curve  line,  and  also  in  column  (i) ;  then 
multiply  the  number  in  (i)  by  the  root  figure,  thus  giving  the  square 
of  the  first  root  figure,  and  write  the  result  in  (n) ;  multiply  the 
number  in  (n)  by  the  root  figure,  thus  giving  the  cube  of  the  first 
root  figure,  and  write  the  result  below  the  first  or  left-hand  period 
in  (in) ;  subtract  it  therefrom,  and  annex  the  next  period  to  the 
remainder  for  a  dividend. 

3.  In  (i)  write  the  root  figure  below  the  former,  and  multiply 
the  sum  of  these  by  the  root  figure ;  place  the  product  in  (n),  and 
add  the  two  numbers  together  for  a  trial  divisor.     Again,  write  the 
root  figure  in  (i),  and  add  it  to  the  former  sum. 

4.  With  the  number  in  (n)  as  a  trial  divisor  of  the  dividend, 
omitting  the  two  figures  to  the  right  of  it,  find  the  next  figure  of 
the  root,  and  annex  it  to  the  former,  and  also  to  the  number  in  (i). 
Multiply  the  number  now  in  (i)  by  the  new  figure  of  the  root,  and 
write  the  product  as  it  arises  in  (n),  but  extended  two  places  of 
figures  more  to  the  right,  and  the  sum  of  these  two  numbers  will 
be  the  corrected  divisor ;  then  multiply  the  corrected  divisor  by  the 


TO  EXTRACT  THE  CUBE  HOOT. 


last  root  figure,  placing  the  product  as  it  arises  below  the  dividend ; 
subtract  it  therefrom,  annex  another  period,  and  proceed  precisely 
as  described  in  (3),  for  correcting  the  columns  (i)  and  (n).  Then 
with  the  new  trial  divisor  in  (n),  and  the  new  dividend  in  (in), 
proceed  as  before. 

When  the  trial  divisor  is  not  contained  in  the  dividend,  after  two 
figures  are  omitted  on  the  right,  the  next  root  figure  is  0,  and  there- 
fore one  cipher  must  be  annexed  to  the  number  in  (i) ;  two  ciphers 
to  the  number  in  (n) ;  and  another  period  to  the  dividend  in  (in). 

When  the  root  is  interminable,  we  may  contract  the  work  very 
considerably,  after  obtaining  a  few  figures  in  the  decimal  part  of 
the  root,  if  we  omit  to  annex  another  period  to  the  remainder  in 
(m) ;  cut  off  one  figure  from  the  right  of  (n),  and  two  figures  from 
(i),  which  will  evidently  have  the  elect  of  cutting  off  three  figures 
from  each  column ;  and  then  work  with  the  numbers  on  the  left,  as 
in  contracted  multiplication  and  division  of  decimals. 

Find  the  cube  root  of  21035-8  to  ten  places  of  decimals. 

(i)  (ii)  .      .     (in) 

2  4  21035-8(27-60491055944 

L  8  8 

4  12..  13035 

2  469  11683 


67 

_7 
74 

JT_ 
816 

_6 
822 

_6_ 

82804 
4^ 

82808 
4 

:8[28|12 


1669 
518 

2 1 8  r . . 

4896 
223596 

4932     . 

228528.  .  .  . 

331216 

2285611216 

331232 


228594244 
7453 


228601697 
7453 


228609151 
83 


2286092314 

83 


1352800 
1341576 

11224  

9142444864 

2081555136 

2057415281 

24139855 

22860923 

1278932 

1143046 

135886 

114305 

21581 

20575 

1006 

914 

"92 

91 


2|2|8|6|0|9|3|2  1 

Required  the  cube  roots  of  the  following  numbers  : — 


48228544,  46656,  and  15069223. 
64481-201,  and  28991029248. 
12821119155125,  and  -000076765625. 
ttf#,  and  16. 
91 J,  and  7f. 

3 


364,  36,  and  247. 
40-1,  and  3072. 
23405,  and  -0425. 
If,  and  2-519842. 
4-5,  and  1-98802366. 


84  THE   PRACTICAL   MODEL    CALCULATOR. 

TO  EXTRACT  ANY  ROOT  WHATEVER. 

LET  N  be  the  given  power  or  number,  n  the  index  of  the  power, 
A  the  assumed  power,  r  its  root,  R  the  required  root  of  N. 

Then,  as  the  sum  of  n  +  1  times  A  and  n  —  1  times  N,  is  to 
the  sum  of  n  -J-  1  times  N  and  n  —  1  times  A,  so  is  the  assumed 
root  r,  to  the  required  root  R. 

Or,  as  half  the  said  sum  of  n  +  1  times  A  and  n  —  1  times  N, 
is  to  the  difference  between  the  given  and  assumed  powers,  so  is  the 
assumed  root  r,  to  the  difference  between  the  true  and  assumed 
roots  ;  which  difference,  added  or  subtracted,  as  the  case  requires, 
gives  the  true  root  nearly. 


Or,  (n  +  1)  .  JA  +  (n  -  1)  .  JN  :  A  w  N  :  :  r  :  R  w  r. 
And  the  operation  may  be  repeated  as  often  as  we  please,  by 
using  always  the  last  found  root  for  the  assumed  root,  and  its  nth 
power  for  the  assumed  power  A. 

To  extract  the  5th  root  0/21035-8. 

Here  it  appears  that  the  5th  root  is  between  7*3  and  7*4.    Taking 
7-3,  its  5th  power  is  20730-71593.     Hence  then  we  have, 

N  =  21035-8;  r  =  7'3;  n  =  5;  $  .  (n  +  1)  =  3;  Un  -  1)  =  2. 
A  =  20730-716 
N  -  A  =  305-084 


A  =  20730-716  N  =  21035-8 

3        2 

8  A  =  62192.148         42071-6 
=  42071*6 
As  104263-7  :  305-084  : :  7-3  :  -0213605 


915252 
2135588 

104263-7)  2227-1132  (    -0213605,  the  difference. 
14184       7-3  =  r  add 
3758 

630        7-321360  =  R,  the  root,  true  to 
5  the  last  figure. 

The  6th  root  of  21035.8  is  5-254037. 

The  6th  root  of  2  is  1-122462. 

The  7th  root  of  21035-8  is  4-145392. 

The  7th  root  of  2  is  1-104089. 

The  9th  root  of  2  is  1-080059. 

OF  RATIOS,  PROPORTIONS,  AND  PROGRESSIONS. 

NUMBERS  are  compared  to  each  other  in  two  different  ways  :  the 
one  comparison  considers  the  difference  of  the  two  numbers,  and 
is  named  Arithmetical  Relation,  and  the  difference  sometimes 
Arithmetical  Ratio :  the  other  considers  their  quotient,  and  is  called 


ARITHMETICAL   PROPORTION   AND   PROGRESSION.  35 

Geometrical  Relation,  and  the  quotient  the  Geometrical  Ratio.  So, 
of  these  two  numbers  6  and  3,  the  difference  or  arithmetical  ratio 
is  6  —  3  or  3 ;  but  the  geometrical  ratio  is  f  or  2. 

There  must  be  two  numbers  to  form  a  comparison :  the  number 
which  is  compared,  being  placed  first,  is  called  the  Antecedent ; 
and  that  to  which  it  is  compared  the  Consequent.  So,  in  the 
two  numbers  above,  6  is  the  antecedent,  and  3  is  the  consequent. 

If  two  or  more  couplets  of  numbers  have  equal  ratios,  or  equal 
differences,  the  equality  is  named  Proportion,  and  the  terms  of  the 
ratios  Proportionals.  So,  the  two  couplets,  4,  2  and  8,  6  are  arith- 
metical proportionals,  because  4  —  2  =  8  —  6  =  2;  and  the  two  cou- 
plets 4,  2  and  6,  3  are  geometrical  proportionals,  because  |  =  f  =  2, 
the  same  ratio. 

To  denote  numbers  as  being  geometrically  proportional,  a  colon 
is  set  between  the  terms  of  each  couplet  to  denote  their  ratio ;  and 
a  double  colon,  or  else  a  mark  of  equality  between  the  couplets  or 
ratios.  So,  the  four  proportionals,  4, 2, 6, 3,  are  set  thus,  4 :  2  : :  6  :  3, 
which  means  that  4  is  to  2  as  6  is  to  3 ;  or  thus,  4:2  =  6:3;  or 
thus,  |  =  |,  both  which  mean  that  the  ratio  of  4  to  2  is  equal  to 
the  ratio  of  6  to  3. 

Proportion  is  distinguished  into  Continued  and  Discontinued. 
When  the  difference  or  ratio  of  the  consequent  of  one  couplet  and 
the  antecedent  of  the  next  couplet  is  not  the  same  as  the  Common 
difference  or  ratio  of  the  couplets,  the  proportion  is  discontinued. 
So,  4,  2,  8,  6  are  in  discontinued  arithmetical  proportion,  because 
4  —  2  =  8  —  6  =  2,  whereas,  2  —  8  =  —  6 ;  and  4,  2,  6,  3  are  in 
discontinued  geometrical  proportion,  because  J  =  | \  =  2,  but  f  =  J, 
which  is  not  the  same. 

But  when  the  difference  or  ratio  of  every  two  succeeding  terms  is 
the  same  quantity,  the  proportion  is  said  to  be  continued,  and  the  num- 
bers themselves  a  series  of  continued  proportionals,  or  a  progression. 
So,  2,  4,  6,  8  form  an  arithmetical  progression,  because  4—2=6— 
4  =  8  —  6  =  2,  all  the  same  common  difference ;  and  2,  4,  8, 16,  a 
geometrical  progression,  because  |  =  f  =  ^8  =  2,  all  the  same  ratio. 

When  the  following  terms  of  a  Progression  exceed  each  other, 
it  is  called  an  Ascending  Progression  or  Series ;  but  if  the  terms 
decrease,  it  is  a  Descending  one. 

So,      0, 1,  2,  3,    4,  &c.,  is  an  ascending  arithmetical  progression, 

but      9,  7,  5,  3,    1,  &c.,  is  a  descending  arithmetical  progression : 

Also,  1,  2,  4,  8, 16,  &c.,  is  an  ascending  geometrical  progression, 

and    16,  8, 4,  2,    1,  &c.,  is  a  descending  geometrical  progression. 

ARITHMETICAL  PROPORTION  AND   PROGRESSION. 

THE  first  and  last  terms  of  a  Progression  are  called  the  Extremes ; 
and  the  other  terms  lying  between  them,  the  Means. 

The  most  useful  part  of  arithmetical  proportions  is  contained  in 
the  following  theorems : 

THEOREM  1. — If  four  quantities  be-  in  arithmetical  proportion,  the 
sum  of  the  two  extremes  will  be  equal  to  the  sum  of  the  two  means. 

Thus,  of  the  four  2,  4,  6,  8,  here  24-8  =  4  +  6=10. 


6b  THE  PRACTICAL  MODEL  CALCULATOR. 

THEOREM  2. — In  any  continued  arithmetical  progression,  the  sum 
of  the  two  extremes  is  equal  to  the  sum  of  any  two  means  that  are 
equally  distant  from  them,  or  equal  to  double  the  middle  term  when 
there  is  an  uneven  number  of  terms. 

Thus,  in  the  terms  1,  3,  5,  it  is  1  +  5  =  3  -f  3  =  6. 

And  in  the  series  2,  4,  6,  8, 10, 12, 14,  it  is  2  -f  14  =  4  +  12  = 
6  +  10  =  8  +  8  =  16. 

THEOREM  3. — The  difference  between  the  extreme  terms  of  an 
arithmetical  progression,  is  equal  to  the  common  difference  of  the 
series  multiplied  by  one  less  than  the  number  of  the  terms. 

So,  of  the  ten  terms,  2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  the  com- 
mon difference  is  2,  and  one  less  than  the  number  of  terms  9  ;  then 
the  difference  of  the  extremes  is  20  —  2  =  18,  and  2  X  9  =  18  also. 

Consequently,  the  greatest  term  is  equal  to  the  least  term  added 
to  the  product  of  the  common  difference  multiplied  by  1  less  than 
the  number  of  terms. 

THEOREM  4. — The  sum  of  all  the  terms  of  any  arithmetical  pro- 
gression is  equal  to  the  sum  of  the  two  extremes  multiplied  by  the 
number  of  terms,  and  divided  by  2 ;  or  the  sum  of  the  two  extremes 
multiplied  by  the  number  of  the  terms  gives  double  the  sum  of  all 
the  terms  in  the  series. 

This  is  made  evident  by  setting  the  terms  of  the  series  in  an 
inverted  order  under  the  same  series  in  a  direct  order,  and  adding 
the  corresponding  terms  together  in  that  order.  Thus, 

in  the  series,       1 ,      3  ,      5  ,      7 ,      9  ,    11 ,    13  ,     15  ; 

inverted,  15  ,    13  ,    11 ,      9  ,      7  ,      5  ,      3  ,      1 ; 

the  sums  are,    16  +  16  -f  16  +  16  +  16  +  16  +  16  +  16, 
which  must  be  double  the  sum  of  the  single  series,  and  is  equal  to 
the  sum  of  the  extremes  repeated  so  often  as  are  the  number  of 
the  terms. 

From  these  theorems  may  readily  be  found  any  one  of  these  five 
parts ;  the  two  extremes,  the  number  of  terms,  the  common  differ- 
ence, and  the  sum  of  all  the  terms,  when  any  three  of  them  are 
given,  as  in  the  following  Problems : 

PROBLEM   I. 

Given  the  extremes  and  the  number  of  terms,  to  find  the  sum  of  all 

the  terms. 

RULE. — Add  the  extremes  together,  multiply  the  sum  by  the 
number  of  terms,  and  divide  by  2. 

The  extremes  being  3  and  19,  and  the  number  of  terms  9 ; 
required  the  sum  of  the  terms  ? 
19 

'          -5|    Or>li+lx9  =  fx9  =  llx9  =  99. 

2)198 

99  =  the  sum. 


ARITHMETICAL   PROPORTION  AND   PROGRESSION.  37 

The  strokes  a  clock  strikes  in  one  whole  revolution  of  the  index, 
or  in  12  hours,  is  78. 

PROBLEM  II. 

G-iven  the  extremes,  and  the  number  of  terms  ;  to  find  the  common 

difference. 

RULE. — Subtract  the  less  extreme  from  the  greater,  and  divide 
the  remainder  by  l>less  than  the  number  of  terms,  for  the  common 
difference. 

The  extremes  being  3  and  19,  and  the  number  of  terms  9 ;  re- 
quired the  common  difference  ? 
19 
_3         .     19-3      16 

8)16       Or>  -9Trr==T==2' 

_2 

If  the  extremes  be  10  and  70,  and  the  number  of  terms  21 ;  what 
is  the  common  difference,  and  the  sum  of  the  series  ? 

The  com.  diff.  is  3,  and  the  sum  is  840. 

PROBLEM   III. 

Griven  one  of  the  extremes,  the  common  difference,  and  the  number 
of  terms;  to  find  the  other  extreme,  and  the  sum  of  the  series. 
RULE. — Multiply  the  common  difference  by  1  less  than  the  num- 
ber of  terms,  and  the  product  will  be  the  difference  of  the  extremes : 
therefore  add  the  product  to  the  less  extreme,  to  give  the  greater ; 
or  subtract  it  from  the  greater,  to  give  the  less. 

Given  the  least  term  3,  the  common  difference  2,  of  an  arith- 
metical series  of  9  terms ;  to  find  the  greatest  term,  and  the  sum 
of  the  series  ? 

2 

_8 

16 

3 

19  the  greatest  term. 
__3  the  least. 
22  sum. 

9  number  of  terms. 
2)198 

99  the  sum  of  the  series. 

If  the  greatest  term  be  70,  the  common  difference  3,  and  the 
number  of  terms  21 ;  what  is  the  least  term  and  the  sum  of  the 
series  ?  The  least  term  is  10,  and  the  sum  is  840. 

PROBLEM   IV. 

To  find  an  arithmetical  mean  proportional  between  two  given  terms. 

RULE. — Add  the  two  given  extremes  or  terms  together,  and  take 

half  their  sum  for  the  arithmetical  mean  required.     Or,  subtract 


38  THE  PRACTICAL  MODEL  CALCULATOR. 

the  less  extreme  from  the  greater,  and  half  the  remainder  will  be 
the  common  difference  ;  which,  being  added  to  the  less  extreme,  or 
subtracted  from  the  greater,  will  give  the  mean  required. 

To  find  an  arithmetical  mean  between  the  two  numbers  4  and  14. 
Here,  14  Or,  14  Or,  14 

_4  _4  _5 

2)18  2)JLO  J9i 

9  5  the  com.  dif. 

4  the  less  extreme. 


So  that  9  is  the  mean  required  by  both  methods. 

PROBLEM  v. 

To  find  two  arithmetical  means  between  two  given  extremes. 
RULE.  —  Subtract  the  less  extreme  from  the  greater,  and  divide 
the  difference  by  3,  so  will  the  quotient  be  the  common  difference  ; 
which,  being  continually  added  to  the  less  extreme,  or  taken  from 
the  greater,  gives  the  means. 

To  find  two  arithmetical  means  between  2  and  8. 

Here  8 

Then  2  -f  2  =  4  the  one  mean, 

and  4  -f  2  =  6  the  other  mean. 
com.  dif.    2 

PROBLEM  VI. 

To  find  any  number  of  arithmetical  means  between  two  given  terms 

or  extremes. 

RULE.  —  Subtract  the  less  extreme  from  the  greater,  and  divide 
the  difference  by  1  more  than  the  number  of  means  required  to  be 
found,  which  will  give  the  common  difference  ;  then  this  being 
added  continually  to  the  least  term,  or  subtracted  from  the  greatest, 
will  give  the  mean  terms  required. 

To  find  five  arithmetical  means  between  2  and  14. 

Here  14 

_     Then,  by  adding  this  com.  dif.  continually, 
6  )^2         the  means  are  found,  4,  6,  8,  10,  12. 
com.  dif.    2 

GEOMETRICAL  PROPORTION  AND  PROGRESSION. 

THE  most  useful  part  of  Geometrical  Proportion  is  contained  in 
the  following  theorems  : 

THEOREM  1.  —  If  four  quantities  be  in  geometrical  proportion, 
the  product  of  the  two  extremes  will  be  equal  to  the  product  of  the 
two  means. 

Thus,  in  the  four  2,  4,  3,  6  it  is  2  X  6  =  3  x  4  =  12. 

And  hence,  if  the  product  of  the  two  means  be  divided  by  one 
of  the  extremes,  the  quotient  will  give  the  other  extreme.  So,  of 


GEOMETRICAL   PROPORTION   AND   PROGRESSION.  39 

the  above  numbers,  the  product  of  the  means  12  -5-  2  =  6  the  one 
extreme,  and  12  -*-  6  =  2  the  other  extreme;  and  this  is  the 
foundation  and  reason  of  the  practice  in  the  Rule  of  Three. 

THEOREM  2. — In  any  continued  geometrical  progression,  the  pro- 
duct of  the  two  extremes  is  equal  to  the  product  of  any  two  means 
that  are  equally  distant  from  them,  or  equal  to  the  square  of  the 
middle  term  when  there  is  an  uneven  number  of  terms. 

Thus,  in  the  terms  2,  4,  8,  it  is  2  x  8  =  4  X  4  =  16. 
And  in  the  series  2,  4,  8,  16,  32,  64,  128, 

it  is  2  x  128  =  4  x  64  =  8  x  32  =  16  x  16  =  256. 
THEOREM  3. — The  quotient  of  the  extreme  terms  of  a  geome- 
trical progression  is  equal  to  the  common  ratio  of  the  series  raised 
to  the  power  denoted  by  one  less  than  the  number  of  the  terms. 

So,  of  the  ten  terms  2,  4,  8,  16,  32,  64,  128,  256,  512,  1024, 
the  common  ratio  is  2,  one  less  than  the  number  of  terms  9 ;  then 

1024 

the  quotient  of  the  extremes  is  — ~ —  =  512,  and  29  =  512  also. 

Consequently,  the  greatest  term  is  equal  to  the  least  term  multi- 
plied by  the  said  power  of  the  ratio  whose  index  is  one  less  than 
the  number  of  terms. 

THEOREM  4. — The  sum  of  all  the  terms  of  any  geometrical  pro- 
gression is  found  by  adding  the  greatest  term  to  the  difference  of 
the  extremes  divided  by  one  less  than  the  ratio. 

So,  the  sum  2,  4,  8, 16,  32,  64, 128,  256,  512, 1024,  (whose  ratio 

1024  —  2 
is  2,)  is  1024  +     2  _  1     =  1024  +  1022  =  2046. 

The  foregoing,  and  several  other  properties  of  geometrical  pro- 
portion, are  demonstrated  more  at  large  in  Byrne's  Doctrine  of  Pro- 
portion. A  few  examples  may  here  be  added  to  the  theorems  just 
delivered,  with  some  problems  concerning  mean  proportionals. 

The  least  of  ten  terms  in  geometrical  progression  being  1,  and 

the  ratio  2,  what  is  the  greatest  term,  and  the  sum  of  all  the  terms  ? 

The  greatest  term  is  512,  and  the  sum  1023. 

PROBLEM  I. 

To  find  one  geometrical  mean  proportional  between  any  two  numbers. 

KULE. — Multiply  the  two  numbers  together,  and  extract  the  square 
root  of  the  product,  which  will  give  the  mean  proportional  sought. 

Or,  divide  the  greater  term  by  the  less,  and  extract  the  square 
root  of  the  quotient,  which  will  give  the  common  ratio  of  the  three 
terms :  then  multiply  the  less  term  by  the  ratio,  or  divide  the 
greater  term  by  it,  either  of  these  will  give  the  middle  term  required. 

To  find  a  geometrical  mean  between  the  two  numbers  3  and  12. 

First  way.  Second  way. 

12  3 )  12  ( 4,  its  root,  is  2,  the  ratio. 

_3 

36  ( 6  the  mean.  Then,  3  x  2  =  6  the  mean. 

36  Or,    12  -T-  2  =  6  also. 


UNIVERSITY 

^  Or-  ^l 


40  THE   PRACTICAL   MODEL   CALCULATOR. 

PROBLEM   II. 

To  find  two  geometrical  mean  proportionals  between  any  two  numbers. 
RULE. — Divide  the  greater  number  by  the  less,  and  extract  the 
cube  root  of  the  quotient,  which  will  give  the  common  ratio  of  the 
terms.     Then  multiply  the  least  given  term  by  the  ratio  for  the 
first  mean,  and  this  mean  again  by  the  ratio  for  the  second  mean ; 
or,  divide  the  greater  of  the  two  given  terms  by  the  ratio  for  the 
greater  mean,  and  divide  this  again  by  the  ratio  for  the  less  mean. 
To  find  two  geometrical  mean  proportionals  between  3  and  24. 
Here,    3  )  24  (  8,  its  cube  root,  2  is  the  ratio. 
Then,    3x2=    6,  and    6  x  2  =  12,  the  two  means. 
Or,      24  -4-  2  =  12,  and  12  -h  2  =    6,  the  same. 

That  is,  the  two  means  between  3  and  24,  are  6  and  12. 

PROBLEM   III. 

To  find  any  number  of  geometrical  mean  proportionals  between  two 

numbers. 

RULE. — Divide  the  greater  number  by  the  less,  and  extract  such 
root  of  the  quotient  whose  index  is  one  more  than  the  number  of 
means  required,  that  is,  the  2d  root  for  1  mean,  the  3d  root  for 
2  means,  the  4th  root  for  3  means,  and  so  on ;  and  that  root  will 
be  the  common  ratio  of  all  the  terms.  Then  with  the  ratio  multi- 
ply continually  from  the  first  term,  or  divide  continually  from  the 
last  or  greatest  term. 

To  find  four  geometrical  mean  proportionals  between  3  and  96. 

Here,     3 )  96  (  32,  the  5th  root  of  which  is  2,  the  ratio. 
Then,     3  x  2=  6,  and   6  x  2=12,  and  12  x  2=24,  and  24x2=48. 
Or,      96  H- 2=48,  and  48 -r-2=24,  and  24n-2=12,  and  12-^-2=  6. 
That  is,  6,  12,  24,  48  are  the  four  means  between  3  and  96. 

OF   MUSICAL   PROPORTION. 

THERE  is  also  a  third  kind  of  proportion,  called  Musical,  which, 
being  but  of  little  or  no  common  use,  a  very  short  account  of  it  may 
here  suffice. 

Musical  proportion  is  when,  of  three  numbers,  the  first  has  the 
same  proportion  to  the  third,  as  the  difference  between  the  first  and 
second  has  to  the  difference  between  the  second  and  third. 

As  in  these  three,  6,  8,  12 ; 

where,   6  :  12  : :  8  -  6  :  12  -  8, 

that  is,  6  :  12  : :  2    :   4. 

"When  four  numbers  are  in  Musical  Proportion ;  then  the  first 
has  the  same  proportion  to  the  fourth,  as  the  difference  between 
the  first  and  second  has  to  the  difference  between  the  third  and 
fourth. 

As  in  these,  6,  8,  12,  18 ; 

where,  6  :  18  : :  8  —  6  :  18  -  12, 

thatis,6  :  18  ::  2    :    6. 


FELLOWSHIP.  41 

When  numbers  are  in  Musical  Progression,  their  reciprocals  are 
in  Arithmetical  Progression ;  and  the  converse,  that  is,  when  num- 
bers are  in  Arithmetical  Progression,  their  reciprocals  are  in  Mu- 
sical Progression. 

So,  in  these  Musicals  6,  8,  12,  their  reciprocals  J,  J,  ^,  are  in 
arithmetical  progression ;  for  ^  +  f%  =  T32  =  J ;  and  |  +  J  =  f  =  f ; 
that  is,  the  sum  of  the  extremes  is  equal  to  double  the  mean,  which 
is  the  property  of  arithmeticals. 

FELLOWSHIP,  OR  PARTNERSHIP. 

FELLOWSHIP  is  a  rule  by  which  any  sum  or  quantity  may  be 
divided  into  any  number  of  parts,  which  shall  be  in  any  given  pro- 
portion to  one  another. 

By  this  rule  are  adjusted  the  gains,  or  losses,  or  charges  of  part- 
ners in  company ;  or  the  effects  of  bankrupts,  or  legacies  in  case  of 
a  deficiency  of  assets  or  effects ;  or  the  shares  of  prizes,  or  the 
numbers  of  men  to  form  certain  detachments ;  or  the  division  of 
waste  lands  among  a  number  of  proprietors. 

Fellowship  is  either  Single  or  Double.  It  is  Single,  when  the 
shares  or  portions  are  to  be  proportional  each  to  one  single  given 
number  only ;  as  when  the  stocks  of  partners  are  all  employed  for 
the  same  time  :  and  Double,  when  each  portion  is  to  be  proportional 
to  two  or  more  numbers ;  as  when  the  stocks  of  partners  are  em- 
ployed for  different  times. 

SINGLE   FELLOWSHIP. 

GENERAL  RULE. — Add  together  the  numbers  that  denote  the 
proportion  of  the  shares.  Then, 

As  the  sum  of  the  said  proportional  numbers 
Is  to  the  whole  sum  to  be  parted  or  divided, 
So  is  each  several  proportional  number 
To  the  corresponding  share  or  part. 
Or,  As  the  whole  stock  is  to  the  whole  gain  or  loss, 

So  is  each  man's  particular  stock  to  his  particular  share  of 
the  gain  or  loss. 

To  prove  the  work. — Add  all  the  shares  or  parts  together,  and 
the  sum  will  be  equal  to  the  whole  number  to  be  shared,  when  the 
work  is  right. 

To  divide  the  number  240  into  three  such  parts,  as  shall  be  in 
proportion  to  each  other  as  the  three  numbers,  1,  2,  and  8. 

Here  1  +  2  +  3  =  6  the  sum  of  the  proportional  numbers. 


Then,  as  6  :  240 
and,  as  6  :  240 
also  as  6  :  240 


40  the  1st  part, 
80  the  2d  part, 
120  the  3d  part. 


Sum  of  all  240,  the  proof, 

Three  persons,  A,  B,  C,  freighted  a  ship  with  340  tuns  of  wine ; 
of  which,  A  loaded  110  tuns,  B  97,  and  C  the  rest :  in  a  storm,  the 


D2 


42  THE  PKACTICAL  MODEL  CALCULATOR. 

seamen  were  obliged  to  throw  overboard  85  tuns ;  now  much  must 
each  person  sustain  of  the  loss  ? 

Here,    110  +    97  =  207  tuns,  loaded  by  A  and  B ; 
theref.,  340  -  207  =  133  tuns,  loaded  by  C. 
hence,  as  340  :  85  : :  110 
or,     as      4:1::  110  :  27^  tuns  =  A's  loss  ; 
and,  as      4  :    1  : :     97  :  24J  tuns  =  B's  loss ; 
also,  as      4:1::  133  :  33|  tuns  =  C's  loss. 
Sum  85   tuns,  the  proof. 

DOUBLE  FELLOWSHIP. 

DOUBLE  FELLOWSHIP,  as  has  been  said,  is  concerned  in  cases 
in  which  the  stocks  of  partners  are  employed  or  continued  for  dif- 
ferent times. 

RULE. — Multiply  each  person's  stock  by  the  time  of  its  continu- 
ance ;  then  divide  the  quantity,  as  in  Single  Fellowship,  into  shares 
in  proportion  to  these  products,  by  saying : 

As  the  total  sum  of  all  the  said  products 

Is  to  the  whole  gain  or  loss,  or  quantity  to  be  parted, 

So  is  each  particular  product 

To  the  corresponding  share  of  the  gain  or  loss. 

SIMPLE  INTEREST. 

INTEREST  is  the  premium  or  sum  allowed  for  the  loan,  or  for- 
bearance of  money. 

The  money  lent,  or  forborne,  is  called  the  Principal. 
The  sum  of  the  principal  and  its  interest,  added  together,  is 
called  the  Amount. 

Interest  is  allowed  at  so  much  per  cent,  per  annum,  which  pre- 
mium per  cent,  per  annum,  or  interest  of  a  $100  for  a  year,  is 
called  the  Rate  of  Interest.  So, 

When  interest  is  at  3  per  cent,  the  rate  is  3 ; 

4  per  cent 4; 

5  per  cent 5; 

6  per  cent 6. 

Interest  is  of  two  sorts :  Simple  and  Compound. 

Simple  Interest  is  that  which  is  allowed  for  the  principal  lent  or 
forborne  only,  for  the  whole  time  of  forbearance. 

As  the  interest  of  any  sum,  for  any  time,  is  directly  proportional 
to  the  principal  sum,  and  also  to  the  time  of  continuance ;  hence 
arises  the  following  general  rule  of  calculation. 

GENERAL  RULE. — As  $100  is  to  the  rate  of  interest,  so  is  any 
given  principal  to  its  interest  for  one  year.  And  again, 

As  one  year  is  to  any  given  time,  so  is  the  interest  for  a  year  just 
found  to  the  interest  of  the  given  sum  for  that  time. 

Otherwise. — Take  the  interest  of  one  dollar  for  a  year,  which, 
multiply  by  the  given  principal,  and  this  product  again  by  the  time 


POSITION.  43 

of  loan  or  forbearance,  in  years  and  parts,  for  the  interest  of  the 
proposed  sum  for  that  time. 

When  there  are  certain  parts  or  years  in  the  time,  as  quarters, 
or  months,  or  days,  they  may  be  worked  for  either  by  taking  the 
aliquot,  or  like  parts  of  the  interest  of  a  year,  or  by  the  Rule  of 
Three,  in  the  usual  way.  Also,  to  divide  by  100,  is  done  by  only 
pointing  off  two  figures  for  decimals. 

COMPOUND   INTEREST. 

COMPOUND  INTEREST,  called  also  Interest  upon  Interest,  is  that 
which  arises  from  the  principal  and  interest,  taken  together,  as  it 
becomes  due  at  the  end  of  each  stated  time  of  payment. 

RULES. — 1.  Find  the  amount  of  the  given  principal,  for  the  time 
of  the  first  payment,  by  Simple  Interest.  Then  consider  this 
amount  as  a  new  principal  for  the  second  payment,  whose  amount 
calculate  as  before  ;  and  so  on,  through  all  the  payments  to  the  last, 
always  accounting  the  last  amount  as  a  new  principal  for  the  next 
payment.  The  reason  of  which  is  evident  from  the  definition  of 
Compound  Interest.  Or  else, 

2.  Find  the  amount  of  one  dollar  for  the  time  of  the  first  pay- 
ment, and  raise  or  involve  it  to  the  power  whose  index  is  denoted 
by  the  number  of  payments.  Then  that  power  multiplied  by  the 
given  principal  will  produce  the  whole  amount.  From  which  the 
said  principal  being  subtracted,  leaves  the  Compound  Interest  of 
the  same ;  as  is  evident  from  the  first  rule. 

POSITION. 

POSITION  is  a  method  of  performing  certain  questions  which  can- 
not be  resolved  by  the  common  direct  rules.  It  is  sometimes  called 
False  Position,  or  False  Supposition,  because  it  makes  a  supposi- 
tion of  false  numbers  to  work  with,  the  same  as  if  they  were  the 
true  ones,  and  by  their  means  discovers  the  true  numbers  sought. 
It  is  sometimes  also  called  Trial  and  Error,  because  it  proceeds 
by  trials  of  false  numbers,  and  thence  finds  out  the  true  ones  by  a 
comparison  of  the  errors. 

Position  is  either  Single  or  Double. 

SINGLE   POSITION. 

SINGLE  POSITION  is  that  by  which  a  question  is  resolved  by  means 
of  one  supposition  only. 

Questions  which  have  their  results  proportional  to  their  supposi- 
tions belong  to  Single  Position ;  such  as  those  which  require  the 
multiplication  or  division  of  the  number  sought  by  any  proposed 
number ;  or,  when  it  is  to  be  increased  or  diminished  by  itself,  or 
any  parts  of  itself,  a  certain  proposed  number  of  times. 

RULE. — Take  or  assume  any  number  for  that  required,  and  per- 
form the  same  operations  with  it  as  are  described  or  performed  in 
the  question. 

Then  say,  as  the  result  of  the  said  operation  is  to  the  position 


44  THE  PRACTICAL  MODEL  CALCULATOR. 

or  number  assumed,  so  is  the  result  in  the  question  to  the  number 
sought. 

A  person,  after  spending  J-  and  \  of  his  money,  has  yet  remain- 
ing $60,  what  had  he  at  first  ? 

Suppose  he  had  at  first  $120  Proof. 

Now  j-  of  120  is    40  J.  of  144  is    48 

J  of  it  is      _30  J  of  144  is  _36 

their  sum  is            70  their  sum      84 

which  taken  from  120  taken  from  144 

leaves                     50  leaves            60  as  per  question. 

Then,  50  :  120  : :  60  :  144. 

What  number  is  that,  which  multiplied  by  7,  and  the  product 
divided  by  6,  the  quotient  may  be  14  ?  12. 

PERMUTATIONS  AND  COMBINATIONS. 

THE  Permutations  of  any  number  of  quantities  signify  the  changes 
which  these  quantities  may  undergo  with  respect  to  their  order. 

Thus,  if  we  take  the  quantities  a,  b,  c ;  then,  a  b  c,  a  c  6,  b  a  c, 
b  c  a,  c  a  b,  c  b  a,  are  the  permutations  of  these  three  quantities 
taken  all  together;  a  b,  a  <?,  b  a,  b  c,  c  a,  c  b,  are  the  permutations 
of  these  quantities  taken  two  and  two;  a,  b,  c,  are  the  permutation 
of  these  quantities  taken  singly,  or  one  and  one,  &c. 

The  number  of  the  permutations  of  the  eight  letters,  «,  6,  c,  d, 
e)  f>  9t  h)  ig  40320  ;  becomes, 

1.2.3.4.5.6.7.8  =  40320. 

DOUBLE  POSITION. 

DOUBLE  POSITION  is  the  method  of  resolving  certain  questions 
by  means  of  two  suppositions  of  false  numbers. 

To  the  Double  Rule  of  Position  belong  such  questions  as  have 
their  results  not  proportional  to  their  positions :  such  are  those,  in 
which  the  numbers  sought,  or  their  parts,  or  their  multiples,  are 
increased  or  diminished  by  some  given  absolute  number,  which  is 
no  known  part  of  the  number  sought. 

Take  or  assume  any  two  convenient  numbers,  and  proceed  with 
each  of  them  separately,  according  to  the  conditions  of  the  ques- 
tion, as  in  Single  Position ;  and  find  how  much  each  result  is  dif- 
ferent from  the  result  mentioned  in  the  question,  noting  also 
whether  the  results  'are  too  great  or  too  little. 

Then  multiply  each  of  the  said  errors  by  the  contrary  supposi- 
tion, namely,  the  first  position  by  the  second  error,  and  the  second 
position  by  the  first  error. 

If  the  errors  are  alike,  divide  the  difference  of  the  products  by 
the  difference  of  the  errors,  and  the  quotient  will  be  the  answer. 

But  if  the  errors  are  unlike,  divide  the  sum  of  the  products  by 
the  sum  of  the  errors,  for  the  answer. 

The  errors  are  said  to  be  alike,  when  they  are  either  both  too 
great,  or  both  too  little  ;  and  unlike,  when  one  is  too  great  and  the 
other  too  little. 


MENSURATION   OF   SUPERFICIES.  45 

What  number  is  that,  which,  being  multiplied  by  6,  the  product 
increased  by  18,  and  the  sum  divided  by  9,  the  quotient  shall  be  20. 
Suppose  the  two  numbers,  18  and  30.     Then 

First  position.                   Second  position.  Proof. 

18                                         30  27 

_6  mult.                                  6  _6 

108                                       180  162 

18  add.                                 18  18 

9)126                                   9)198  9)180 

14  results.                             22  20 
20  true  res.                           20 
+  6  errors  unlike.                —  2 
2d  pos._30  mult.                                18  1st  pos. 

Errors  f2  ™>  ~^ 


Sum        8 )  216  sum  of  products. 
27  answer  sought. 

Find,  by  trial,  two  numbers,  as  near  the  true  number  as  possible, 
and  operate  with  them  as  in  the  question ;  marking  the  errors 
which  arise  from  each  of  them. 

Multiply  the  difference  of  the  two  numbers,  found  by  trial,  by 
the  least  error,  and  divide  the  product  by  the  difference  of  the 
errors,  when  they  are  alike,  but  by  their  sum  when  they  are  unlike. 

Add  the  quotient,  last  found,  to  the  number  belonging  to  the 
least  error,  when  that  number  is  too  little,  but  subtract  it  when  too 
great,  and  the  result  will  give  the  true  quantity  sought. 

MENSURATION  OF  SUPERFICIES. 

THE  area  of  any  figure  is  the  measure  of  its  surface,  or  the  space 
contained  within  the  bounds  of  that  surface,  without  any  regard  to 
thickness. 

A  square  whose  side  is  one  inch,  one  foot,  or  one  yard,  &c.  is 
called  the  measuring  unity  and  the  area  or  content  of  any  figure  is 
computed  by  the  number  of  those  squares  contained  in  that  figure. 

To  find  the  area  of  a  parallelogram;  whether  it  be  a  square,  a 
rectangle,  a  rhombus,  or  a  rhomboides. — Multiply  the  length  by  the 
perpendicular  height,  and  the  product  will  be  the  area. 

The  perpendicular  height  of  the  parallelogram  is  equal  to  the 
area  divided  by  the  base. 

Required  the  area  of  the  square  ABCD  whose   — ; 
side  is  5  feet  9  inches. 

Here  5ft.  9  in.  =  5-75 :  and~^J5\2  =  5-75  X 
5-75  =  33-0625  feet  =  BBfe.  0  in.  9 pa.  =  area 
required. 


46 


THE  PRACTICAL  MODEL  CALCULATOR. 


Required  the  area  of  the  rectangle 
ABCD,  whose  length  AB  is  13-75  chains, 
and  breadth  BC  9-5  chains. 

Here  13-75   X  9-5  =  130-625;  and 

=  13-0625  ac.  =  13  ac.  0  ro.  10     A 


po.  =  area  required. 

Required  the  area  of  the  rhombus 
ABCD,  whose  length  AB  is  12  feet  6 
inches,  and  its  height  DE  9  feet  3  inches. 

Here  12  fe.  6  in.  =  12-5,  and  9  fe.  3  in. 

=  9-25. 

Whence,  12-5  x  9-25  =  115-625  fe.  — 
115  fe.  7  in.  6  pa.  =  area  required. 

What  is  the  area  of  the  rhom- 
boides  ABCD,  whose  length  AB  is 
10-52  chains,  and  height  DE  7-63 
chains. 

Here  10-52  X  7'63  =  80-2676 ; 


and 


=  8-02676  acres  =  8  ac. 


0  ro.  4po.  area  required. 

To  find  the  area  of  a  triangle. — Multiply  the  base  by  the   per- 
pendicular height,  and  half  the  product  will  be  the  area. 

The  perpendicular  height  of  the  triangle  is  equal  to  twice  the 
area  divided  by  the  base. 

Required  the  area  of  the  triangle  ABC, 
whose  base  AB  is  10  feet  9  inches,  and 
height  DC  7  feet  3  inches. 

Here  10  fe.  9  in.  =  10-75,  and  7fe.  3  in. 
=-  7-25. 

Whence,  10-75  x  7*25  =  77*9375,  and 

V?j}JL.L  =  38-96875  feet  =  38  fe.  11  in.     A  D       B 

71  pa.  =  area  required. 

To  find  the  area  of  a  triangle  whose  three  sides  only  are  given. — 
From  half  the  sum  of  the  three  sides  subtract  each  side  severally. 

Multiply  the  half  sum  and  the  three  remainders  continually  toge- 
ther, and  the  square  root  of  the  product  will  be  the  area  required 

Required  the  area  of  the  triangle  ABC,  c 

whose  three  sides  BC,  CA,  and  AB  are 
24,  36,  and  48  chains  respectively. 

24  -f  36  +  48       108 
Mere ^ ^  ~o~   =  54  = 

J  sum  of  the  sides. 

Also,  54  -  24  =  BO  first  diff. ;  54  -  36 
=  18  second  diff.;  and  54  —  48  =  6  third  diff. 


MENSURATION   OF   SUPERFICIES. 


47 


418-282  =  area 


Whence,  V  54  X  30  X  18  X  6  = 
required. 

Any  two  sides  of  a  right  angled  triangle  being  given  to  find 
the  third  side. — When  the  two  legs  are  given  to  find  the  hypo- 
thenuse, add  the  square  of  one  of  the  legs  to  the  square  of  the 
other,  and  the  square  root  of  the  sum  will  be  equal  to  the  hypo- 
thenuse. 

When  the  hypothenuse  and  one  of  the  legs  are  given  to  find  the 
other  leg. — From  the  square  of  the  hypothenuse  take  the  square 
of  the  given  leg,  and  the  square  root  of  the  remainder  will  be  equal 
to  the  other  leg. 

In  the  right  angled  triangle  ABC,  the  c 

base  AB  is  56,  and  the  perpendicular  BC  38, 
what  is  the  hypothenuse  ? 

Here  562  +  332  =  3136  +  1089  =  4225, 
and  \/4225  =  65  =  hypothenuse  AC. 

If  the  hypothenuse  AC  be  53,  and  the 
base  AB  45,  what  is  the  perpendicular  BC  ?     A 

Here  532  -  452  =  2809  -  2025  =  784,  and  v/784  =  28  = 
perpendicular  BC. 

To  find  the  area  of  a  trapezium. — Multiply  the  diagonal  by  the 
sum  of  the  two  perpendiculars  falling  upon  it  from  the  opposite 
angles,  and  half  the  product  will  be  the  area. 

Required  the  area  of  the  trapezium 
BAED,  whose  diagonal  BE  is  84,  the 
perpendicular  AC  21,  and  DF  28. 

Here  28  +  21  x  84  =49  X  84=4116, 
4116 

and     2     =  2058  the  area  required. 

To  find  the  area  of  a  trapezoid,  or  a  quadrangle,  two  of  whose 
opposite  sides  are  parallel. — Multiply  the  sum  of  the  parallel  sides 
by  the  perpendicular  distance  between  them,  and  half  the  product 
will  be  the  area. 

Required  the  area  of  the  trapezoid  ABCD, 
whose  sides  AB  and  DC  are  321-51  and 
214-24,  and  perpendicular  DE  171-16. 

Here  321-51  +  214-24  =  535-75  =  sum 
of  the  parallel  sides  AB,  DC. 

Whence,  535-75  X  171-16  (theperp.  DE)  = 

91698-9700,  and  -   )—^-     -  =  45849-485  the  area  required. 

To  find  the  area  of  a  regular  polygon. — Multiply  half  the  peri- 
meter of  the  figure  by  the  perpendicular  falling  from  its  centre 
upon  one  of  the  sides,  and  the  product  will  be  the  area. 

The  perimeter  of  any  figure  is  the  sum  of  all  its  sides. 


48 


THE  PRACTICAL  MODEL  CALCULATOR. 


Required  the  area  of  the  regular  pentagon 
ABODE,  whose  side  AB,  or  BC,  &c.,  is  25 
feet,  and  the  perpendicular  OP  17 '2  feet. 

or    y     r 

Here  — Q —  =  62-5  =  half  perimeter, 

and  62-5  X  17*2  =  1075  square  feet  =  area 

required. 

A        r        is 

To  find  the  area  of  a  regular  polygon,  when  the  side  only  is 
given. — Multiply  the  square  of  the  side  of  the  polygon  by  the 
number  standing  opposite  to  its  name  in  the  following  table,  and 
the  product  will  be  the  area. 


No.  of 

sides. 

Names. 

Multipliers. 

No.  of 
sides. 

Names. 

Multipliers. 

3 
1  4 
5 
6 

7 

Trigon  or  equil.  A 
Tetragon  or  square 
Pentagon 
Hexagon 
Heptagon 

0-433013 
1-000000 
1-720477 
2-598076 
3-633912 

8 
9 
10 
11 
12 

Octagon 
Nonagon 
Decagon 
Undecagon 
Duodecagon 

4-828427 
6-181824 
7-694209 
9-365640 
11-196152 

The  angle  OBP,  together  with  its  tangent,  for  any  polygon  of  not 
more  than  12  sides,  is  shown  in  the  following  table : 


No.  of 
sides. 

Names. 

Angle 
OBP. 

Tangents. 

3 
4 

5 
6 

7 
8 
9 

10 
11 
12 

Trigon 
Tetragon 

Pentagon 
Hexagon 
Heptagon 
Octagon 
Nonagon 

Decagon 
Undecagon 
Duodecagon 

30° 
45° 

54° 
60° 
64°f 
67°£ 
70° 

72° 
73<YT 
75° 

•57735  =  £  v/3 
1-00000  =  1X1 

1-37638  =  v/T+T\/5 
1-73205  =  v/3 
2-07652 
2-41421  =  1  +  ^/2 
2-74747 

3-07768  =  v'S  +  2  ^5 
3-40568 
3-73205  =  2  +  v/3 

Required  the  area  of  a  pentagon  whose  side  is  15. 

The  number  opposite  pentagon  in  the  table  is  1*720477. 

Hence  1*720477  x  152  =  1*720477  x  225  =  387*107325  = 
area  required. 

The  diameter  of  a  circle  being  given  to  find  the  circumference, 
or  the  circumference  being  given  to  find  the  diameter. — Multiply 
the  diameter  by  3-1416,  and  the  product  will  be  the  circumfer- 
ence, or 

Divide  the  circumference  by  3*1416,  and  the  quotient  will  be  the 
diameter. 

As  7  is  to  22,  so  is  the  diameter  to  the  circumference ;  or  as  22 
is  to  7,  so  is  the  circumference  to  the  diameter. 

As  113  is  to  355,  so  is  the  diameter  to  the  circumference ;  or, 
as  352  is  to  115,  so  is  the  circumference  to  the  diameter. 


MENSURATION   OF   SUPERFICIES.  49 

If  the  diameter  of  a  circle  be  17,  what  is  the  circumference  ? 

Here  3-1416  X  17  =  534072  =  circumference. 
If  the  circumference  of  a  circle  be  354,  what  is  the  diameter  ? 

354-000 
Here  '  =  112-681  =  diameter. 


To  find  the  length  of  any  arc  of  a  circle. — When  the  chord  of 
the  arc  and  the  versed  sine  of  half  the  arc  are  given  : 

To  15  times  the  square  of  the  chord,  add  33  times  the  square  of 
the  versed  sine,  and  reserve  the  number. 

To  the  square  of  the  chord,  add  4  times  the  square  of  the  versed  sine, 
and  the  square  root  of  the  sum  will  be  twice  the  chord  of  half  the  arc. 

Multiply  twice  the  chord  of  half  the  arc  by  10  times  the  square 
of  the  versed  sine,  divide  the  product  by  the  reserved  number,  and 
add  the  quotient  to  twice  the  chord  of  half  the  arc :  the  sum  will 
be  the  length  of  the  arc  very  nearly. 

When  the  chord  of  the  arc,  and  the  chord  of  half  the  arc  are 
given. — From  the  square  of  the  chord  of  half  the  arc  subtract  the 
square  of  half  the  chord  of  the  arc,  the  remainder  will  be  the  square 
of  the  versed  sine :  then  proceed  as  above. 

When  the  diameter  and  the  versed  sine  of  half  the  arc  are  given  : 

From  60  times  the  diameter  subtract  27  times  the  versed  sine, 
and  reserve  the  number. 

Multiply  the  diameter  by  the  versed  sine,  and  the  square  root 
of  the  product  will  be  the  chord  of  half  the  arc. 

Multiply  twice  the  chord  of  half  the  arc  by  10  times  the  versed 
sine,  divide  the  product  by  the  reserved  number,  and  add  the  quo- 
tient to  twice  the  chord  of  half  the  arc ;  the  sum  will  be  the  length 
of  the  arc  very  nearly. 

When  the  diameter  and  chord  of  the  arc  are  given,  the  versed 
sine  may  be  found  thus  :  From  the  square  of  the  diameter  subtract 
the  square  of  the  chord,  and  extract  the  square  root  of  the  re- 
mainder. Subtract  this  root  from  the  diameter,  and  half  the  re- 
mainder will  give  the  versed  sine  of  half  the  arc. 

The  square  of  the  chord  of  half  the  arc  being  divided  by  the 
diameter  will  give  the  versed  sine,  or  being  divided  by  the  versed 
sine  will  give  the  diameter. 

The  length  of  the  arc  may  also  be  found  by  multiplying  together  the 
number  of  degrees  it  contains,  the  radius  and  the  number  -01745329. 

Or,  as  180  is  to  the  number  of  degrees  in  the  arc,  so  is  3-1416 
times  the  radius,  to  the  length  of  the  arc. 

Or,  as  3  is  to  the  number  of  degrees  in  the  arc,  so  is  -05236  times 
the  radius  to  the  length  of  the  arc.  0 

If  the  chord  DE  be  48,  and  the  versed  sine      D. 
CB  18,  what  is  the  length  of  the  arc  ?  / 

Here  482  x  15  =  34560  ! 

182  x  33  =  10692 

45252  reserved  number. 


50  THE  PRACTICAL  MODEL  CALCULATOR. 

482  =  2304  =  the  square  of  the  chord. 
182  x  4  =  1296  =  4  times  the  square  of  the  versed  sine. 
^/  3600"  =  60  =  twice  the  chord  of  half  the  arc. 
60  x  182  x  10       194400 
w 4525? ~  45059   =  4 '2 959,  which  added  to  twice 

the  chord  of  half  the  arc  gives  64*2959  =  the  length  of  the  arc. 

50  X  60  =  3000 
18  x  27  =    486 

2514  reserved  number. 


AC  =  N/50  x  18  =  30  =  the  chord  of  half  the  are. 


30  x  2  x  18  x  10       10800        A  ••.-       7  .  7      77  7 

=ss:   0^14    s=s  4*2959,  which  added  to  twice  the 


chord  of  half  the  arc  gives  64-2959  =  the  length  of  the  arc. 

To  find  the  area  of  a  circle.  —  Multiply  half  the  circumference  by 
half  the  diameter,  and  the  product  will  be  the  area. 

Or  take  J  of  the  product  of  the  whole  circumference  and  diameter. 
What  is  the  area  of  a  circle  whose  diameter  is  42,  and  circum- 
ference 131*946  ? 

2)131*946 

65*973  =  J  circumference. 
21  =  i  diameter. 

65973 
131946 


1385*433  =  area  required. 

What  is  the  area  of  a  circle  whose  diameter  is  10  feet  6  inches, 
and  circumference  31  feet  6  inches  ? 
fe.        in. 

15         9  =  15*75  =  J  circumference. 
5         3  =    5*25  =  \  diameter. 
7875 
3150 
7875 
82*6875 

12 

8*2500 

82  feet  8  inches. 

Multiply  the  square  of  the  diameter  by  -7854,  and  the  product 
will  be  the  area ;  or, 

Multiply  the  square  of  the  circumference  by  '07958,  and  the 
product  will  be  the  area. 

The  following  table  will  also  show  most  of  the  useful  problems 
relating  to  the  circle  and  its  equal  or  inscribed  square. 

Diameter  X  -8862  =  side  of  an  equal  square. 

Circumf.    X  -2821  =  side  of  an  equal  square. 

Diameter  X  -7071  =  side  of  the  inscribed  square. 


MENSURATION   OF   SUPERFICIES.  51 

Circumf.  X  -2251  =  side  of  the  inscribed  square. 
Area  X  -6366  =  side  of  the  inscribed  square. 
Side  of  a  square  X  1*4142  =  diam.  of  its  circums.  circle. 
Side  of  a  square  X  4-443  =  circumf.  of  its  circums.  circle. 
Side  of  a  square  X  1*128  =  diameter  of  an  equal  circle. 
Side  of  a  square  X  3 '545  =  circumf.  of  an  equal  circle. 
What  is  the  area  of  a  circle  whose  diameter  is  5  ? 
7854 

25  =  square  of  the  diameter. 
39270 
15708 
19-6350  =  the  answer. 

To  find  the  area  of  a  sector,  or  that  part  of  a  circle  which  is 
bounded  by  any  two  radii  and  their  included  arc. — Find  the  length 
of  the  arc,  then  multiply  the  radius,  or  half  the  diameter,  by  the 
length  of  the  arc  of  the  sector,  and  half  the  product  will  be  the 
area. 

If  the  diameter  or  radius  is  not  given,  add  the  square  of  half  the 
chord  of  the  arc,  to  the  square  of  the  versed  sine  of  half  the  arc ; 
this  sum  being  divided  by  the  versed  sine,  will  give  the  diameter. 

The  radius  AB  is  40,  and  the  chord  BC 
of  the  whole  arc  50,  required  the  area  of  _j> 

the  sector. 

Cf^ ! ^B 

80  -  ^780*  -  502 

^ =  8*7750  =  the  versed 

sine  of  half  the  arc. 


80  x  60  —  8-7750  x  27  =  4563-0750 

the  reserved  number. 


2  x  v/8-7750  x  80  =  52*9906  =  twice  the 
chord  of  half  the  arc. 

52-9906  x  8*7750  x  10 

— 4553.0750 —       ~"  =  1*0190,  which  added  to  twice  the  chord 

of  half  the  arc  gives  54*0096  the  length  of  the  arc. 

54*0096  x  40 
And  -     — o —      ~  =  1080*1920  =  area  of  the  sector  required. 

As  360  is  to  the  degrees  in  the  arc  of  a  sector,  so  is  the  area  of 
the  whole  circle,  whose  radius  is  equal  to  that  of  the  sector,  to  the 
area  of  the  sector  required. 

For  a  semicircle,  a  quadrant,  &c.  take  one  half,  one  quarter,  &c. 
of  the  whole  area. 

The  radius  of  a  sector  of  a  circle  is  20,  and  the  degrees  in  its 
arc  22 ;  what  is  the  area  of  the  sector  ? 

Here  the  diameter  is  40. 

Hence,  the  area  of  the  circle  =  402  X  *7854  =  1600  X  *7854  = 
1256*64 

Now,  360°  :  22°  : :  1256*64  :  76*7947  =  area  of  the  sector. 


52  THE   PRACTICAL   MODEL   CALCULATOR. 

To  find  the  area  of  a  segment  of  a  circle. — Find  the  area  of 
the  sector,  having  the  same  arc  with  the  segment,  by  the  last  pro- 
blem. 

Find  the  area  of  the  triangle  formed  by  the  chord  of  the  seg- 
ment, and  the  radii  of  the  sector. 

Then  the  sum,  or  difference,  of  these  areas,  according  as  the 
segment  is  greater  or  less  than  a  semicircle,  will  be  the  area  re- 
quired. 

The  difference  between  the  versed  sine  and  radius,  multiplied  by 
half  the  chord  of  the  arc,  will  give  the  area  of  the  triangle. 

The  radius  OB  is  10,  and  the  chord  AC  10  ; 
what  is  the  area  of  the  segment  ABC  ? 


TJIT  =  ~20~  —  5  =  the  versed  sine 


AC2       100 

of  half  the  arc. 

20  x  60  -  5  X  27  =  1065  =  the  reserved 
number. 

2  : 

1065 —      7  =  "9390,  and  this  added 

to  twice  the  chord  of  half  the  arc  gives  20-9390  =  the  length  of  the  arc. 

20-9390  X  10 

— o—      -  =  104-6950  =  area  of  the  sector  OACB. 

OD  =  OC  =  CD  =  5  the  perpendicular  height  of  the  triangle. 
AD  =  x/AO2  —  OD2  =  v/75  =  8-6603  =  J  the  chord  of  the  arc. 
8-6603  X  5  =  43-3015  =  the  area  of  the  triangle  AOB. 
104-6950  -  43-3015  =  61-3935  =  area  of  the  segment  required; 
it  being  in  this  case  less  than  a  semicircle. 

Divide  the  height,  or  versed  sine,  by  the  diameter,  and  find  the 
quotient  in  the  table  of  versed  sines. 

Multiply  the  number  on  the  right  hand  of  the  versed  sine  by  the 
square  of  the  diameter,  and  the  product  will  be  the  area. 

When  the  quotient  arising  from  the  versed  sine  divided  by  the 
diameter,  has  a  remainder  or  fraction  after  the  third  place  of  deci- 
mals ;  having  taken  the  area  answering  to  the  first  three  figures, 
subtract  it  from  the  next  following  area,  multiply  the  remainder  by 
the  said  fraction,  and  add  the  product  to  the  first  area,  then  the 
sum  will  be  the  area  for  the  whole  quotient. 

If  the  chord  of  a  circular  segment  be  40,  its  versed  sine  10,  and 
the  diameter  of  the  circle  50,  what  is  the  area  ? 
5-0)1-0 

•2  =  tabular  versed  sine. 
•111823  =  tabular  segment. 

2500  =  square  of  50. 
55911500 
223646 
279-557500  =  area  required. 


MENSURATION   OF   SUPERFICIES. 


53 


To  find  the  area  of  a  circular  zone,  or  the  space  included  between 
any  two  parallel  chords  and  their  intercepted  arcs. — From  the 
greater  chord  subtract  half  the  difference  between  the  two,  mul- 
tiply the  remainder  by  the  said  half  difference,  divide  the  product 
by  the  breadth  of  the  zone,  and  add  the  quotient  to  the  breadth. 
To  the  square  of  this  number  add  the  square  of  the  less  chord,  and 
the  square  root  of  the  sum  will  be  the  diameter  of  the  circle. 

Now,  having  the  diameter  EG,  and  the  two  chords  AB  and  DC, 
find  the  areas  of  the  segments  ABEA,  and  DCED,  the  difference 
of  which  will  be  the  area  of  the  zone  required. 

The  difference  of  the  tabular  segments  multiplied  by  the  square 
of  the  circle's  diameter  will  give  the  area  of  the  zone. 

When  the  larger  segment  AEB  is  greater  than  a  semicircle,  find 
the  areas  of  the  segments  AGB,  and  DCE,  and  subtract  their  sum 
from  the  area  of  the  whole  circle  :  the  remainder  will  be  the  area 
of  the  zone. 

The  greater  chord  AB  is  20,  the  less  DC  15, 
and  their  distance  Dr  17  J  :  required  the  area 
of  the  zone  ABCD. 

20  —  15 

2 =  2-5  =  J  =  the  difference  between 

the  chords. 

IT.U  J.  (2Q  ~  2-5)  x  2-5 

1  *7  •  £\ 

20  =  DF. 


17-5  +  2-5  = 


And  v/202  +  152  =  v/625  =  25  =  the  diameter  of  the  circle. 

The  segment  AEB  being  greater  than  a  semicircle,  we  find  the 
versed  sine  o/DCE  =  2-5,  and  that  0/AGB  =  5. 

2-5 
Hence  -^r  =  '100  =  tabular  versed  sine  of  DEC. 

And  or  =  -200  =  tabular  versed  sine  of  AGB. 

Now  -040875  x  252  =  area  of  seg.  DEC  =  25-546875 
And  -111823  x  252  =  area  of  seg.  AGB  =  69-889375 

sum  95-43625 

•7854  X  252  =  area  of  the  whole  circle,      =  490-87500 
Difference  =  area  of  the  zone   ABCD  =  395-43875 

To  find  the  area  of  a  circular  ring,  or  the 
space  included  between  the  circumference  of 
two  concentric  circles. — The  difference  between 
the  areas  of  the  two  circles  will  be  the  area  of 
the  ring. 

Or,  multiply  the  sum  of  diameters  by  their 
difference,  and  this  product  again  by  -7854, 
and  it  will  give  the  area  required. 

The  diameters  AB  -and  CD  are  20  and  15 :  required  the  area  of 


54 


THE   PRACTICAL   MODEL   CALCULATOR. 


the  circular  ring,   or  the  space  included  between  the  circumfe- 
rences of  those  circles. 


Here  AB  +  CD  X  AB  —  CD  =  35  X  5  =  175, andVlb  X  -7854  == 
137*4450  =  area  of  the  ring  required. 

To  find  the  areas  of  lunes,  or  the  spaces  between  the  intersecting 
arcs  of  two  eccentric  circles. — Find  the  areas  of  the  two  segments 
from  which  the  lune  is  formed,  and  their  difference  will  be  the 
area  required. 

The  following  property  is  one  of  the  most  curious : 

If  ABC  be  a  right  angled  triangle, 
and  semicircles  be  described  on  the  three 
sides  as  diameters,  then  will  the  said  tri- 
angle be  equal  to  the  two  lunes  D  and  F 
taken  together. 

For  the  semicircles  described  on  AC  and       

BC  =  the  one  described  on  AB,  from  each       ^  ~~B 

take  the  segments  cut  off  by  AC  and  BC,  then  will  the  lunes  AFCE 
and  BDCGr  =  the  triangle  ACB. 

The  length  of  the  chord  AB  is  40,  the 
height  DC  10,  and  DE  4:  required  the 
area  of  the  lune  ACBEA. 

The  diameter  of  the  circle  of  which  ACB 

202  +  102 
is  a  part  = JQ —  =  50. 

And  the  diameter  of  the  circle  of  which  AEB  is  apart 
=  104. 

Now  having  the  diameter  and  versed  sines,  we  find, 

The  area  of  seg.  ACB  =  -111823  X  502  =  279-5575 
And^  area  of  sea.  AEB  =  -009955  X  1042  =  107*6733 
Their  difference  is  the  area  of  the  lune  1 
AEBCA  required,  / 


171-8842 


To  find  the  area  of  an  irregular  polygon,  or  a  figure  of  any 
number  of  sides. — Divide  the  figure  into  triangles  and  trapeziums, 
and  find  the  area  of  each  separately. 

Add  these  areas  together,  and  the  sum  will  be  equal  to  the  area 
of  the  whole  polygon. 

Required  the  area  of  the  irre- 
gular figure  ABCDEFGA,  the  fol- 
lowing lines  being  given : 

CO  =  6 


GB  =  30-5 
GD  =  29 

FD  =  24-8 


11' 
11 
4 


Here 


An  +  Co 


xGB  = 
x  30-5  = 


11-2  +  6 


2 

x  30-5  +  8-6  x  30-5  =  262-3  = 
area  of  the  trapezium  ABCG. 


DECIMAL   APPROXIMATIONS. 


55 


And 


C* 


„_ 
X  GD 


11  +  6-6 
=  --  --  x 


•8  x  29  =-  255-2  = 


area  of  the  trapezium  GCDF. 

FD  x  E»      24-8  x  4      99-2 
Also, H —  — 9 ^  ~2~—  49-6  =  area  of  the  triangle 

FDE. 

Whence  262-3  +  255-2  +  49-6  =  567-1  =  area  of  the  whole 
fyure  required. 

DECIMAL  APPROXIMATIONS   FOR  FACILITATING   CALCULATIONS   IN 
MENSURATION. 


Lineal  feet  multiplied  by 

•00019 

= 

miles. 

—      yards 

___ 

•000568 

— 

- 

Square  inches 

— 

•007 

= 

square  feet. 

—      yards 

— 

•0002067 

= 

acres. 

Circular  inches 

— 

•00546 

== 

square  feet. 

Cylindrical  inches 

— 

•0004546 

= 

cubic  feet. 

feet 

— 

•02909 

= 

cubic  yards. 

Cubic  inches 

— 

•00058 

= 

cubic  feet. 

—     feet 

— 

•03704 

= 

cubic  yards. 

—      — 

— 

6-232 

= 

imperial  gallons. 

—     inches 

— 

•003607 

= 

— 

Cylindrical  feet 

— 

4-895 

== 

— 

inches 

— 

•002832 

= 

— 

Cubic  inches 

— 

•263 

= 

Ebs.  avs.  of  cast  iron 

— 

— 

•281 

= 

—     wrought  do. 

— 

— 

•283 

= 

—     steel. 

— 

— 

•3225 

= 

—     copper. 

— 

— 

•3037 

= 

—     brass. 

— 

— 

•26 

= 

—     zinc. 

— 

— 

•4103 

== 

—     lead. 

— 

— 

•2636 

= 

—     tin. 

— 

— 

•4908 

= 

—     mercury. 

Cylindrical  inches 

— 

•2065 

= 

—     cast  iron. 

— 

— 

•2168 

= 

—     wrought  iron. 

— 

— 

•2223 

= 

—     steel. 

— 

— 

•2533 

= 

—     copper. 

— 

— 

•2385 

= 

—     brass. 





•2042 

= 

—     zinc. 



— 

•3223 

= 

—     lead. 

— 

— 

•207 

= 

—     tin. 

— 

— 

•3854 

= 

—     mercury. 

Avoirdupois  Bbs. 

— 

•009 

= 

cwts. 

— 

— 

•00045 

= 

tons. 

183-346  circular  inches                         = 

1  square  foot. 

2200  cylindrical  inches                          = 

1  cubic  foot. 

French  metres  X  3-281                         = 

feet. 

—      kilogrammes 

x  2- 

205 

= 

avoirdupois  Ib. 

—      grammes  X  • 

002205                = 

avoirdupois  Ibs. 

56 


THE  PRACTICAL  MODEL  CALCULATOR. 


Diameter  of  a  sphere  X  -806    =  dimensions  of  equal  cube. 

Diameter  of  a  sphere  X  -6667  =  length  of  equal  cylinder. 

Lineal  inches  X  -0000158         =  miles. 

A  French  cubic  foot  =  209347  cubic  inches. 

Imperial  gallons  X  -7977  =  New  York  gallons. 

The  average  quantity  of  water  that  falls  in  rain  and  snow  at 
Philadelphia  is  36  inches. 

At  West  Point  the  variation  of  the  magnetic  needle,  Nov.  16th, 
1839,  was  7°  58'  27"  West,  and  the  dip  73°  26'  28". 


DECIMAL  EQUIVALENTS   TO   FRACTIONAL   PARTS    OF   LINEAL 
MEASURES. 


One  inch,  the  integer  or  whole  number. 

•96875       I  +  A 

•625           | 

•28125       \  +  A 

•9375         |  +  A 

•59375       i  +  ^ 

•25             \ 

•90625        J  +  A 

•5625         I  +  A 

•21875       i  -f  ^ 

•875      3  l 

•53125  3  1  +  A 

•1875    3  14-  A 

•84375  -3  }  +  & 

•5          **  i- 

•15625  -3  1  +  A 

•8125     g<  f  +  Jj 

•46875  &  §  +  A 

•125       &  i 

•78125   «  |  +  A 

•4375     *  |  +  A 

•09375  «          ,| 

•75         £  | 

•40625   S  f  +  A 

•0625     §          A 

•71875       |  +  & 

•375           f 

•03125               A 

•6875         |  +  A 

•34375       I  +  A 

•65625       |  +  A 

•3125         \  +  ^ 

One  foot,  or  12  inches,  the  integer. 

•9166      11  inches. 

•4166         5  inches. 

•0625        f  of  in. 

•6338  ^  10     — 

.3333    -^  4     — 

•05208  *•  f    — 

•75      |    9     — 

•25        |  3     — 

.Q4166  g  I    — 

•6666  §f   8     — 

•1666    ^2     — 

•03125  r§    — 

•5833  £    7     — 

•0833    g  1     — 

•02083  g  1    — 

•5        «    6     — 

•07291  *  1    — 

•01041  «  1    — 

One  yard,  or  36  inches,  the  integer. 

•9722      35  inches. 

•6389       23  inches. 

•3055       11  inches. 

.9444      34     _ 

•6111      22     — 

•2778      10     — 

•9167      33     — 

•5833      21     — 

•25            9     — 

•8889  0  32     — 

•5556  o  20     — 

•2222  o    8     — 

•8611;:  31    — 

•5278;:  19     — 

•1944  ^    7     — 

•8333  §  30     — 

•5        g  18     — 

•1667  §    6     — 

•8056  g  29     — 

•4722  Sfl7     — 

•1389  g"   5     — 

•7778  £  28     — 

•4444  g  16     — 

•1111  g    4     — 

•75      *  27     — 

•4167  *  15     — 

•0833  *     3     — 

•7222      26     — 

•3889      14     — 

•0555        2     — 

•6944      25     — 

•3611      13     — 

•0278        1     — 

•6667      24     — 

•3333      12     — 

Table  containing  the  Circumferences,  Squares,  Cubes,  and  Areas  of 
Circles,  from  1  to  100,  advancing  by  a  tenth. 


Diara 

Circum 

Square 

Cube. 

Area 

Diam. 

Circum. 

Square. 

Cube. 

Area. 

1, 

3-1416 

1 

1 

•7884 

9 

28-2744 

81 

729 

63-6174 

| 

3-4557 

1-21 

1-331 

•9503 

•1 

28-5885 

82-81 

753-571 

65-0389 

•2 

3-7699 

1-44 

1-728 

1-1309 

•2 

28-9027 

84-64 

778-688 

66-4762 

•3 

4-0840 

1-69 

2-197 

1-3273 

•3 

29-2168 

86-49 

804-357 

67-9292 

•4 

43982 

1-96 

2-744 

1-5393 

•4 

29-5310 

88-36 

830-584 

69-3979 

•5 

4-7124 

2-25 

3-375 

1-7671 

•5 

29-8452 

90-25 

857-375 

70-8823 

•6 

5-0265 

2-56 

4-096 

2-0106 

•6 

30-1593 

92-16 

884-736 

72-3824 

•7 

53407 

2-89 

4-913 

2-2698 

•7 

30-4735 

94-09 

912-673 

73-8982 

•8 

5-6548 

3-24 

5-832 

2-5446 

•8 

30-7876 

96-04 

941-192 

75-4298 

•9 

5-9690 

3-61 

6-859 

2-8352 

•9 

31-1018 

98-01 

970-299 

76-9770 

2 

6-2832 

4 

8 

31416 

10 

31-4160 

100 

1000 

78-5400 

•1 

6-5973 

4-41 

9-C61 

3-4636 

•1 

31-7301 

102-01 

1030-301 

80-1186 

•2 

6-9115 

4-84 

10-648 

3-8013 

•2 

320443 

104-04 

1061-208 

81-7130 

•3 

7-2256 

5-29 

12-167 

4-1547 

•3 

32-3580 

106-09 

1092-727 

83-3230 

•4 

7-5398 

5-76 

13-824 

4-5239 

•4 

32-6726 

108-16 

1124-864 

849488 

•5 

7-8540 

6-25 

15-625 

4-9087 

•5 

32-9868 

110-25 

1157-625 

865903 

•6 

8-1681 

6-76 

17-576 

5-3093 

•6 

33-3009 

112-36 

1191-016 

88-2475 

•7 

8-4823 

7-29 

19-683 

5-7255 

•7 

336151 

114-49 

1225-043 

89-9204 

•8 

8-7964 

7-84 

21-952 

6-1575 

•8 

33-9292 

116-64 

1259-712 

91-6090 

•9 

9-1106 

8-41 

24-389 

6-6052 

•9 

34-2434 

118-81 

1295029 

93-3133 

3 

9-4248 

9 

27 

7-0686 

11 

34-5576 

121 

1331 

95-0334 

•1 

9-7389 

9-61 

29-791 

7-5476 

•1 

34-87  17 

123-21 

1367-631 

96-7691 

•2 

10-0531 

10-24 

32-768 

80424 

•2 

35-1859 

125-44 

1404-928 

98-5205 

•3 

10-3672 

10-89 

35-937 

8-5530 

•3 

35-5010 

127-69 

1442-897 

100-2877 

•4 

10-6814 

11-56 

39-304 

9-0792 

•4 

35-8142 

129-96 

1481-544 

102-0705 

•5 

10-9956 

1225 

42-875 

9-6211 

•5 

36-1284 

132-25 

1520-875 

103-8691 

•6 

11  3097 

12-96 

46-656 

10-1787 

•6 

36-4425 

134-56 

1560-896 

105-6834 

•7 

11-6239 

13-69 

50-653 

10-7521 

•7 

36-7567 

136-89 

1601-613 

107-5134 

•8 

11-9380 

1444 

54-872 

11-3411 

•8 

370708 

139-24 

1643-032 

109-3590 

•9 

12-2522 

15-21 

59-319 

11-9459 

•9 

37-3840 

141-61 

1685-159 

111-2204 

4 

12-5664 

16 

64 

12-5664 

12 

37-6992 

144 

1728 

113-0976 

•1 

12-8805 

16-81 

68-921 

132025 

•1 

38-0133 

146-41 

1771-561 

114-9904 

•2 

13-1947 

17-64 

74-088 

13-8544 

•2 

38-3275 

148-84 

1815-848 

116-8989 

•3 

13-5088 

18-49 

79-507 

14-5220 

•3 

38-6416 

151-29 

1860-867 

118-8231 

•4 

13-8230 

19-36 

85-184 

15-2053 

•4 

38-9558 

153-76 

1906-624 

120-7631 

•5 

14-1372 

20-25 

91-125 

15-9043 

•5 

39-2700 

156-25 

1953-125 

122-7187 

•6 

14-4513 

21-16 

97-336 

16-6190 

•6 

39-5841 

158.76 

2000-376 

124-6901 

•7 

14-7655 

22-09 

103-823 

17-3494 

•7 

39-8983 

lfil-29 

2048383 

126-6771 

•8 

15-0796 

23-04 

110-592 

18-0956 

•8 

40-2124 

163-84 

2097-152 

128  6799 

•9 

15-3938 

24-01 

117-649 

18-8574 

•9 

40-5266 

166-41 

2146-689 

130-6984 

5 

15-7080 

25 

125 

19-6350 

13 

40-8408 

169 

2197 

132-7326 

•1 

16-0221 

26-01 

132-651 

20-4282 

•1 

41-1549 

171-61 

2248-091 

1347824 

•2 

16-3363 

27-04 

140-608 

21-2372 

•2 

41-4691 

174-24 

2299-968 

1368480 

•3 

16-6504 

28-09 

148-877 

22-0618 

•3 

41-7832 

176-89 

2352-637 

1389294 

•4 

16-9646 

29-16 

157-464 

22-9022 

•4 

42-0974 

179-56 

2406  104 

141-0264 

•5 

17-2788 

30-25 

166-375 

23-7583 

•5 

42-4116 

182-25 

2460-375 

143-1391 

•6 

17-5929 

31-36 

175-616 

24-6301 

•6 

42-7257 

184-96 

2515-456 

145-2675 

•7 

17-9071 

32-49 

185-193 

25-5176 

•7 

43-0399 

187-69 

2571-353 

147-4117 

•8 

18-2212 

33-64 

195-112 

26-4208 

•8 

43-3540 

190-44 

2628-072 

149  5715 

•9 

18-5354 

34-81 

205-379 

27-3397 

•9 

43-6682 

193-21 

2685-619 

151-7471 

6 

18-8496 

36 

216 

28-2744 

14 

43-9824 

196 

2744 

153-9384 

•1 

19-1637 

37-21 

226-981 

29-2247 

•1 

44-2965 

198-81 

2803-221 

156-1453 

•2 

19-4779 

38-44 

238-328 

30-1907 

•2 

44-6107 

201-64 

2863-288 

158-3680 

•3 

19-7920 

39-69 

250-047 

31-1725 

•3 

44-9248 

204-49 

2924-207 

160-6064 

•4 

20-1062 

40-96 

262-144 

82-1699 

•4 

45-2390 

207-36 

2985-984 

162-8605 

•5 

20-4204 

42-25 

274-625 

33-1831 

•5 

45-5532 

210-25 

3048-625 

165-1303 

•6 

20-7345 

43-56 

287-496 

34-2120 

•6 

458673 

213-16 

3112-136 

167-4158 

•7 

21-0487 

44-89 

300-763 

35-2566 

•7 

46-1815 

216-09 

3176-523 

169-7179 

•8 

21-3628 

46-24 

314-432 

36-3168 

•8 

46-4956 

219-04 

3241-792 

1720340 

•9 

21-6770 

47-61 

328-509 

37-3928 

•9 

46-8098 

222-01 

3307-949 

174-3666 

7 

21-9912 

49 

343 

38-4846 

15 

47-1240 

225 

3375 

176-7150 

•1 

22-3053 

50-41 

357-911 

39-5920 

•1 

47-4381 

228-01 

3442-951 

179-0790 

•2 

22-6195 

61-84 

373-248 

40-7151 

•2 

47-7523 

231-04 

3511-808 

181-4588 

•3 

229336 

53-29 

389-017 

41-8539 

•3 

48-0664 

234-09 

3581-577 

183-8542 

•4 

23-2478 

54-76 

405-224 

430085 

•4 

48-3806 

237-16 

3652-264 

186-2654 

•5 

23-5620 

56-25 

421-875 

44-1787 

•5 

48-6948 

240-25 

3723875 

188-6923 

•6 

23-8761 

57-76 

438-976 

45-3647 

•6 

49-0089 

243-36 

3796-416 

191-1349 

•7 

24-1903 

59-29 

456-533 

46-5663 

•7 

49-3231 

246-49 

3869-893 

193-5932 

•8 

24-5044 

60-84 

474-552 

47-7837 

•8 

49-6372 

249-64 

3944-312 

196-0672 

•9 

24-8186 

62-41 

493-039 

49-0168 

•9 

49-9514 

252-81 

4019-679 

198-5569 

8 

25-1328 

64 

612 

50-2656 

16 

50-2656 

256 

4096 

201-0624 

•1 

26-4469 

65-61 

531-441 

51-5300 

•1 

50-5797 

259-21 

4173-281 

203-5835 

•2 

25-7611 

67-24 

v     551-368 

52-8102 

•2 

50-8939 

262-44 

4251-528 

206-1209 

•3 

26-0752 

68-89 

571-787 

54-1062 

•3 

51-2080 

26569 

4330-747 

208-6723 

•4 

26-3894 

70-S6 

592-704 

55-4178 

•4 

51-5224 

268-96 

4410-944 

211-1411 

•5 

26-7036 

72-25 

614-125 

56-7451 

•5 

51-8364 

272-25 

4492-125 

213-8251 

•6 

27-0177 

73-96 

636-056 

58-0881 

•6 

52-1505 

275-56 

4574-296 

216-4248 

•7 

27-3319 

75-69 

658-503 

59-4469 

•7 

52-4647 

278-89 

4657-463 

219-0402 

•8 

27-6460 

77-44 

681-472 

60-8213 

•8 

52-7788 

282-24 

4741-632 

221-6712 

•9 

27-9602 

79-21 

704-969 

62-2115 

•9 

53-0930 

285-61 

4826-809 

224-3180 

57 


58 


THE    PRACTICAL   MODEL   CALCULATOR. 


Diam. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Square. 

Cube. 

Area, 

17 

53-4072 

289 

4913 

226-9806 

25 

78-5400 

625 

15625 

490-8750 

•1 

53-7213 

292-41 

5000-211 

229-6588 

•1 

78-8541 

630-01 

1581H-251 

494*098 

•2 

54-0355 

295-84 

5088-448 

232-3527 

•2 

79-1683 

635-04 

16003-008 

498-7604 

•3 

54-3496 

299-29 

5177-717 

235-0623 

•3 

79-4824 

640-09 

16194-277 

502-7266 

•4 

54-6038 

302-76 

5268-024 

237-7877 

•4 

79-7966 

645-16 

16387-064 

506-7086 

•5 

64-9780 

306-25 

5359-375 

240-5287 

•5 

80-8108 

650-25 

16581-375 

510-7063 

•6 

55-2921 

309-76 

5451-776 

243-2855 

•6 

80-4249 

655-36 

16777-216 

514-7196 

•7 

55-6063 

313-29 

5545-233 

2460579 

•7 

80-7391 

660-49 

16974-593 

518-7488 

•8 

55-9204 

316-84 

5639-752 

248-8461 

•8 

81-0532 

665-64 

17173-512 

522-7936 

•9 

56-2346 

320-41 

5735-339 

251-6500 

•9 

81-3674 

670-81 

17373-979 

526-8541 

18 

56-5488 

324 

5832 

254-4696 

26 

81-6816 

676 

17576 

530-9304 

•1 

56-8629 

327-61 

5929-741 

257-3048 

•1 

81-9976 

681-21 

17779-581 

535-0223 

•2 

57-1771 

331-24 

6028-568 

260-1558 

•2 

82-3099 

686-44 

17984-728 

539-1299 

•3 

57-4912 

334-89 

6128-487 

263-0226 

•3 

82-6240 

691-69 

18191-447 

543-2533 

•4 

57-8054 

338-56 

6229-504 

265.9050 

•4 

82-9382 

696-96 

18399-744 

547-3923 

•5 

58-1196 

342-25 

633L-625 

268-8031 

•5 

83-2524 

702-25 

18609-625 

551-5471 

•6 

58-4337 

345-96 

6434-856 

271-7169 

•6 

83-5665 

707-56 

18821-096 

555-7176 

•7 

68-7479 

349-69 

6539-203 

274-6465 

•7 

83-8807 

712-89 

19034-163 

559-9038 

•8 

59-0620 

353-44 

6644-672 

277-5917 

•8 

84-1948 

718-24 

19248-832 

564-1056 

9 

59-3762 

357-21 

6751-269 

280-5527 

•9 

84-5090 

723-61 

19465-109 

568-3232 

19 

59-6904 

361 

6859 

283-5294 

27 

84-8232 

729 

19683 

572-5566 

•1 

60-0045 

364-81 

6967-871 

286-5217 

•1 

85-1373 

734-41 

19902-511 

576-8056 

•2 

60-3187 

368-64 

7077-888 

289-5298 

•2 

85-4515 

739-84 

20123-648 

581-0703 

•3 

60-6328 

372.49 

7189-057 

292-5536 

•3 

85-7656 

745-29 

20346-417 

585-3507 

•4 

60-9470 

376.36 

7301-384 

295-5931 

•4 

86-0798 

750-76 

20570824 

589-6469 

•5 

01-2612 

380-25 

7414-875 

298-6483 

•5 

86-3940 

756-25 

20796-875 

593-9587 

•6 

61-5753 

384-16 

7529-536 

301-7192 

•6 

86-7081 

761-76 

21024-576 

598-2863 

•7 

61-8895 

388-09 

7645-373 

304-8060 

•7 

87-0223 

767-29 

21253-933 

602-6295 

•8 

62-2036 

392-04 

7762-392 

307-9082 

•8 

87-3364 

772-84 

21484-952 

606-9885 

•9 

62-5178 

396-01 

7880-599 

311-0252 

•9 

87-6506 

778-41 

21717-639 

611-3632 

20 

62-8320 

400 

8000 

314-1600 

28 

87-9648 

784 

21952 

615-7536 

•1 

63-1461 

404-01 

8120-601 

317-3094 

•1 

88-2789 

789-61 

22188-041 

620-1596 

•2 

63-4603 

40804 

8242-408 

320-4746 

•2 

88-5931 

795-24 

22425-768 

624-5814 

•3 

63-7744 

412-09 

8365-427 

323-6554 

•3 

88-9072 

800-89 

22665-187 

629-0190 

•4 

64-0886 

416-16 

8489-664 

326-8520 

•4 

89-2214 

806-56 

22906-304 

633-4722 

**5 

64-4028 

420-25 

8615-125 

330-0643 

•5   89-5356 

81225 

23149-125 

637-9411 

•6 

64-7161 

424-36 

8741-816 

333-2923 

•6 

89-8497 

817-96 

23393-656 

642-4257 

•7 

65-0311 

428-49 

8869-743 

336-5360 

•7 

90-1639 

823-69 

23639-903 

6-46-9261 

•8 

65-3452 

432-64 

8998-912 

3397954 

•8 

90-4780 

829-44 

23887-872 

651-4421 

•9 

65-6594 

435-81 

9129-329 

343-0705 

•9 

90-7922 

835-21 

24137-569 

655-9739 

21 

65-9736 

441 

9261 

346-3614 

29 

91-1064 

841 

24389 

660-5214 

•1 

66-2870 

445-21 

9393-931 

349-6679 

•1 

91-4205 

846-81 

24642-171 

665-0845 

•2 

66-6012 

449-44 

9528-128 

352-9901 

•2 

91-7347 

852-64 

24897-088 

669-6634 

•3 

66-7916 

453-69 

9663-597 

356-3281 

•3 

92-0488 

858-49 

25153-757 

674-2580 

•4 

67-2930 

457-96 

9800-344 

359-6817 

•4 

92-3630 

864-36 

25412-184 

678-8683 

•5 

67-5444 

462-25 

9938-375 

3630511 

•5 

92-6772 

870-25 

25672-375 

683-4943 

•6 

67-8585 

466-56 

10077-696 

366-4362 

•6 

929913 

876-16 

25934-336 

688-1360 

•7 

68-1727 

470-89 

10218-313 

369-8370 

•7 

93-3055 

882-09 

26198-073 

692-7934 

•8 

68-4868 

475-24 

10360-232 

373-2534 

•8 

93-6196 

888-04 

26403-592 

697-4666 

•9 

68-8010 

479-61 

10503-459 

376-6856 

•9 

93-9338 

894-01 

26730-899 

702-1554 

22 

6iH152 

484 

10648 

380-1336 

30 

94-2480 

900   . 

27000 

706-8600 

•1 

69-4293 

488-41 

10793-861 

383-5972 

•1 

94-5621 

906-01 

27270-901 

711-5802 

•2 

69-7435 

492-84 

10941-048 

387-0765 

•2 

94-8763 

912-04 

27543608 

716-3162 

•3 

70-0576 

497-29 

11089-567 

390-5751 

•3 

95-1904 

918-09 

27818-127 

721-0678 

•4 

70-3718 

501-76 

11239-424 

394-0823 

•4 

95-5046 

924-16 

28094-464 

725-8352 

•5 

70-6860 

506-25 

11390-625 

397-6087 

•5 

95-8188 

930-25 

28372-625 

730-6183 

•6 

710001 

510-76 

11543-176 

401-1509 

•6 

96-1329 

930-36 

28652-616 

735-4171 

•7 

71-3143 

515-29 

11697-083 

404-7087 

"7 

96-4471 

942-49 

28934-443 

740-2316 

•8 

71-6284 

519-84 

11852-352 

408-2823 

•8 

96-7612 

948-64 

29218-112 

745-0618 

•9 

71-9426 

524-41 

12008-989 

411-8716 

•9 

97-0754 

954-81 

29503-629 

749  9077 

23 

72-2568 

529 

12167 

415-4766 

31 

97-3896 

961 

29791 

754-7694 

•1 

72-5709 

533-61 

12326-391 

419-0972 

•1 

97-7037 

967-21 

30080-231 

759-6467 

•2 

72-8851 

538-24 

12487-168 

422-7336 

•2 

98-0179 

973-44 

30371-328 

764-5397 

•3 

73-1992 

542-89 

12649-337 

426-3858 

•3 

98-3320 

979-69 

30664-297 

769-4485 

•4 

73-5134 

547-56 

12812-904 

430-0536 

•4 

98-6452 

985-96 

30959-144 

774-3729 

•5 

73-8276 

552-25 

12977-875 

4337371 

•5 

98-9604 

992  25 

31255-875 

779-3131 

•6 

74-1417 

556-96 

13144-256 

437-4363 

•6 

99-2745 

998-56 

31554-496 

784-2689 

•7 

74-4559 

561-69 

13312-053 

441-1511 

•7 

99-5887 

1004-89 

31855-013 

789-2406 

•8 

74-7680 

666-44 

13481-272 

444-8819 

•8 

99-9028 

1011-24 

32157-432 

794-2278 

•9 

75-0882 

571-21 

13651-919 

448-6283 

•9 

100-2170 

1017-61 

32461-759 

799-2308 

24 

75-3984 

576 

138-24 

452-3904 

32 

100-5312 

1024 

32768 

804-2496 

•1 

75-7125 

580-81 

13997-541 

456-1681 

•1 

100-8453 

1030-41 

33076-161 

809-2840 

•2 

76-0267 

585-64 

14172-488 

459-9616 

•2 

101-1595 

1036-84 

38386-248 

814-3341 

•3 

76-3408 

590-49 

14348-907 

463-7708 

•3 

101-4736 

1043-29 

33698-267 

819-3999 

•4 

76-6525 

59.5-36 

14526-784 

467-5957 

•4 

101-7478 

1049-76 

34012-224 

824-4815 

•5 

76-9692 

600-25 

14706-125 

471-4363 

•5 

102-1020 

1056-25 

34328-125 

829-5787 

•6 

77-2833 

605-16 

14886-936 

475-2926 

•6 

102-4161 

106276 

34645-976 

834-6917 

•7 

77-5975 

610-09 

15069-223 

479-1646 

'7 

102-7303 

1069-29 

34965-783 

839-8203 

•8 

77-9116 

61504 

1^252-992 

483-0524 

•8 

103-0444 

1075-84 

35287-552 

844-9647 

•9 

78-2258 

620-01 

15438-249 

486-9558 

•9 

103-3586 

1082-41 

35611-289 

850-1248 

CIRCLES,   ADVANCING   BY  A   TENTH. 


59 


Diam. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Square. 

Cube. 

Area. 

33 

103-6728 

10S9 

35937 

855-3006 

41 

128-8056 

1681 

68921 

1320-2574 

•1 

103-9869 

1095-61 

36264-691 

860-4920 

•1 

129-1197 

1689-21 

69426-531 

1326-7055 

•2 

104-3011 

1102-24 

36594-368 

865-6992 

•2 

129-4323 

1697-44 

69934-528 

1333-1693 

•3 

104-6151 

1108-89 

36926-037 

870-9222 

•3 

129-7480 

1705-69 

70444-997 

1339-6489 

•4 

104-9294 

1115-56 

37259-704 

876-1608 

•4 

130-0622 

1713-96 

70957-944 

1346-1441 

•5 

105-2436 

1122-25 

37595-375 

881-4151 

•5 

130-3764 

1722-25 

71473-375 

1352-6551 

•6 

105-5577 

1128-96 

37933-056 

886-6851 

•6 

130-6905 

1730-56 

71991-296 

1359-1818 

•7 

105-8719 

1135-69 

38272-753 

891-9709 

•7 

131-0047 

1738-89 

72511-713 

1365-7242 

•8 

106-1860 

1142-44 

38614-472 

897-2723 

•8 

131-3188 

1747-24 

73034-632 

1372-2822 

•9 

106-5002 

1149-21 

38958-219 

902-5895 

•9 

131-6320 

1755-61 

73560-059 

1378-8560 

34 

106-8144 

1156 

39304 

907-9224 

42 

131-9472 

1764 

74088 

1385-4456 

•1 

107-1285 

1162-81 

39651-821 

913-2709 

•1 

132-2613 

1772-41 

74618-461 

1392-0508 

•2 

107-4272 

1169-64 

40001-688 

918-6352 

•2 

132-5755 

1780-84 

75151-448 

1398-6717 

•3 

107-7568 

1176-49 

40353-607 

924-0115 

•3 

132-8896 

1789-29 

75C86-967 

1405-3083 

•4 

108-0710 

1183-36 

40707-584 

929-4109 

•4 

133-2038 

1797-76 

76225-024 

1411-9607 

•5 

108-3852 

1190-25 

41063-625 

934-8223 

•5 

133-5180 

1806-25 

76765-625 

1418-6287 

•6 

108-6993 

1197-16 

41421-736 

940-2494 

•6 

133-8321 

1814-76' 

77308776 

1425-3125 

•7 

109-0352 

1204-09 

41781-923 

945-6922 

•7 

134-1463 

1823-29 

77854-483 

1432-0119 

•8 

109-3076 

1211-04 

42144-192 

951-1508 

•8 

134-4604 

1831-84 

78402-752 

1438-7271 

•9 

109-6418 

1218-01 

42508-549 

956-6250 

•9 

134-7746 

1840-41 

78958-589 

1445-4580 

35 

109-9560 

1225 

42875 

962-1150 

43 

135-0888 

1849 

79507 

1452-2046 

•1 

110-2701 

1232-01 

43243-551 

967-6206 

•1 

135-4029 

1857-61 

80062-991 

1458-9668 

•2 

110-5843 

1239-04 

43614-208 

973-1420 

•2 

135-7171 

1866-24 

80621-568 

1465-7448 

•3 

110-8984 

1246-09 

43986-977 

978-6790 

•3 

136-0332 

1874-89 

81182-737 

1472-5385 

•4 

111-2126 

1253-16 

44361-864 

984-2318 

•4 

136-3454 

1883-56 

81746-504 

1479-3480 

•5 

111-5268 

1260-25 

44738-875 

989-8003 

•5 

136-6596 

1892-25 

82312-875 

1486-1731 

•6 

111-8409 

1267-36 

45118-016 

995-3845 

•6 

136-9737 

1900-96 

82881-856 

1493-0139 

•7 

112-1551 

1274-49 

45499-293 

1000-9843 

•7 

137-2879 

1909-69 

83453-453 

1499-8705 

•8 

112-4692 

1281-64 

45882-712 

1006-6000 

•8 

137-6020 

1918-44 

84027-672 

1506-7427 

•9 

112-7834 

1288-81 

46268-279 

1012-2313 

•9 

137-9162 

1927-21 

84604-519 

1513-C287 

36 

113-0976 

1236 

46656 

1017-8784 

44 

138-2304 

1936 

85184 

1520-5344 

•1 

113-4117 

1303-21 

47045-831 

1023-5411 

•1 

138-5445 

1944-81 

85766-121 

1527-4537 

•2 

113-7259 

1310-44 

47437-928 

1029-2195 

•2 

138-8587 

1953-64 

86350-888 

1534-3888 

•3 

114-0400 

1317-69 

47832-147 

1034-9131 

•3 

139-1728 

1962-49 

86938-307 

1541-3396 

•4 

114-3542 

1324-96 

48228-544 

1040-6235 

•4 

139-4870 

1971-36 

87528-384 

1548-3061 

•5 

114-6684 

1332-25 

48627-125 

1046-3491 

•5 

139-8012 

1980-25 

88121-125 

1555-2883 

•6 

114-9825 

1339-56 

49027-896 

1052-0904 

•6 

140-1153 

1989-16 

88716-536 

1562-2802 

•7 

115-2967 

1346-89 

49430-863 

1057-8474 

•7 

140-4295 

1998-09 

89314-623 

1569-2998 

•8 

115-6108 

1354-24 

49836-032 

1063-6200 

•8 

140-7436 

2007-04 

89915-392 

1576-3292 

•9 

115-9250 

1361-61 

50243-409 

1069-4084 

•9 

141-0578 

201601 

90518-849 

1583-3742 

37 

116-2392 

1369 

50653 

1075-2126 

45 

141-3720 

2025 

91125 

1590-4350 

•1 

116-5533 

1376-41 

51064-811 

1081-0324 

•1 

141-6861 

2034-01 

91733-851 

1597-5114 

•2 

116-8675 

1383-84 

51478-848 

1086-8679 

•2 

142-0003 

2043-04 

92345-408 

]  604-6036 

•3 

117-1816 

1391-29 

51895-117 

1092-7191 

•3 

142-3144 

2052-09 

92959-677 

1611-7114 

•4 

117-4958 

1398-76 

52313-624 

1098-5862 

•4 

142-6286 

2061-16 

93576-664 

1618-8350 

•5 

117-8100 

1406-25 

52734-375 

1104-4687 

•5 

142-9428 

2070-25 

94196-375 

1625-9743 

•6 

118-1241 

1413-76 

53157-376 

1110-3671 

•6 

143-2569 

2079-36 

94818-816 

1633-1293 

•7 

118-4383 

1421-29 

53582-633 

1116-2811 

•7 

143-5711 

2088-49 

95443-993 

1040-3020 

•8 

118-7524 

1428-84 

54010-152 

1122-2109 

•8 

143-8852 

2097-64 

96071-912 

1647-4864 

•9 

119-0666 

1436-41 

54439-939 

1128-1564 

•9 

144-1994 

2106-81 

96702-579 

1654-C885 

38 

119-3808 

1444 

54872 

11341176 

46 

144-5136 

2116 

97336 

1661-9064 

•1 

119-6949 

1451-61 

55306-341 

1140-0946 

•1 

144-8277 

2125-21 

97972-181 

1669-1399 

•2 

120-0091 

1459-24 

55742-968 

1146-0870 

•2 

145-1419 

2134-44 

98611-128 

1676-3891 

•3 

120-3232 

1466-89 

56181-887 

1152-0954 

•3 

145-4560 

2143-69 

99252-847 

1683-6541 

•4 

120-6374 

1474-56 

56623-104 

1158-1194 

•4 

145-7702 

2152-96 

99897-344 

1690-9347 

•5 

120-9516 

1482-25 

57066-625 

1164-1591 

•5 

146-0844 

2162-25 

100544-625 

1698-2311 

•6 

121-2657 

1489-96 

57512-456 

1170-2145 

•6 

146-3985 

2171-56 

101194-696 

1705-5432 

*Y 

121-5799 

1497-69 

57960-603 

1176-2857 

•7 

146-7127 

2180-89 

101847-563 

17128710 

•8 

121-8940 

1505-44 

58411-072 

1182-3725 

•8 

147-0268 

2190-24 

102503-232 

1720-2144 

•9 

122-2082 

1513-21 

58863-869 

1188-4651 

•9 

147-3410 

2199-61 

103161-709 

1727-5736 

39 

122-5224 

1521 

59319 

1294-5394 

47 

147-6552 

2209 

103823 

1734-9486 

•1 

122  8365 

1528-81 

59776-471 

1200-7273 

•1 

147-9693 

2218-41 

104487-111 

1742-3392 

•2 

123-1507 

1536-64 

60236-288 

1206-8770 

•2 

148-2835 

2227-84 

105154-048 

1749-7455 

•3 

123-4648 

1544-49 

60698-457 

12130424 

•3 

148-5976 

2237-29 

105823-817 

1757-1675 

•4 

123-7790 

1552-36 

61162-984 

1219-2243 

•4 

148-9118 

2246-76 

106496-424 

1704-6045 

•5 

124-0932 

1560-25 

61629-875 

1225-4203 

•5 

149-2260 

2256-25 

107171-875 

17720587 

•6 

124-4073 

1568-16 

62099-136 

1231-6328 

•6 

149-5361 

2265-76 

107850-176 

1779-5279 

•7 

124-7215 

1576-09 

62570-773 

1237-8610 

•7 

149-8543 

2275-29 

108531-333 

1787-0127 

•8 

125-0356 

1584-04 

63044-792 

1244-1210 

•8 

150-1684 

2284-84 

109215-352 

1794-5133 

•9 

125-3498 

1592-01 

63521-199 

1250-3646 

•9 

150-4826 

2294-41 

109902-239 

1802-0296 

40 

125-6640 

1600 

64000 

1256-6400 

48 

150-7968 

2304 

110592 

1809-5616 

•1 

125-9781 

1608-01 

64481-201 

1262-9310 

•1 

151-1109 

2313-61 

111284-641 

1817-1092 

•2 

126-2923 

1616-04 

64964-808 

1269-2888 

•2 

151-4251 

2323-24 

111980-168 

1824-6726 

•3 

126-6064 

1624-09 

65450-827 

1275-5602 

•3 

151-7392 

2332-89 

112678-587 

1832-2518 

•4 

126-9206 

1632-16 

65939-264 

1281-8984 

•4 

152-0534 

2342-56 

113379-904 

1839-8466 

•5 

127-2348 

1640-25 

66430-125 

1288-2523 

•5 

152-3676 

2352-25 

114084-125 

1847-4571 

•6 

127-5489 

1648-36 

66923-416 

1294-6219 

•6 

152-6817 

2361-96 

114791-256 

1855-0833 

•7 

127-8631 

1656-49 

67419143 

1301-0071 

•7 

152-9959 

2371-69 

115501-303 

1862-7253 

•8 

128-1772 

1664-64 

67917-312 

1307-4082 

•8 

153-3100 

2381-44 

116214-272 

1870-3829 

•9 

128-4914 

1672-81 

68417-929 

1313-8249 

•9 

153-6242 

2391-21 

116930-169 

1878-0563 

THE   PRACTICAL   MODEL    CALCULATOR. 


Diam. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Square. 

Cube. 

Area. 

49 

153-9384 

2401 

117649 

1885-7454 

57 

179-0712 

3249 

185193 

2551-7P46 

•1 

154-2525 

2410-81 

118370-771 

1893-4501 

•1 

179-3853 

3260-41 

1S6163-411 

25CO-72CO 

•2 

154-5667 

2420-64 

119095-488 

1901-1706 

•2 

179-6995 

3271-84 

187149-248 

25C9-7031 

•3 

154-8808 

2430-49 

119823-157 

19U8-9U68 

•3 

180-0136 

3283-29 

188132-517 

2578-6959 

•4 

155-1950 

2440-36 

120553-784 

1916-6587 

•4 

180-3278 

3294-76 

189119-224 

2587-7045 

•5 

155-5092 

2450-25 

121287-375 

1924-4263 

•5 

180-6420 

3306-25 

190109-375 

2596-7287 

•6 

155-8233 

2460-16 

122023-936 

1932-2096 

•6 

180-9561 

3317-76 

191102-976 

2605-7687 

.7 

156-1375 

2470-09 

122763-473 

1940-0086 

•7 

181-2803 

3329-29 

192100-033 

2614-8243 

•8 

156-4516 

2480-04 

123505-992 

1947-8234 

•8 

181-5844 

3340-84 

193100-552 

2623-8957 

•9 

156-7558 

2490-01 

124251-499 

1955-6538 

•9 

181-8986 

3352-41 

194104-539 

2632-9828 

50 

157-0800 

2500 

125000 

1963-5000 

58 

182-2128 

3364 

195112 

2642-0856 

•1 

157-3941 

2510-01 

125751-501 

1971-3618 

•1 

182-5269 

3375-61 

196122-941 

2651-2046 

•2 

157-7083 

2520-04 

126506-008 

1979-2394 

•2 

1828411 

3387-24 

197137-368 

2660-3G82 

•3 

158-0224 

2530-09 

127263-527 

1987-1326 

•3 

183-1552 

3398-89 

198155-287 

2669-4882 

•4 

158-3366 

2540-16 

128024-064 

1995-0416 

•4 

183-4694 

3410-56 

199176-704 

2678-6538 

•5 

158-6508 

2550-25 

128787-625 

20029663 

•5 

183-7836 

3422-25 

200201-625 

2687-8351 

•6 

158-9649 

2560-36 

129554-216 

2010-9067 

•6 

184-0977 

3433-96 

201230056 

2697-0321 

•7 

159-2791 

2570-49 

130323-843 

2018-8628 

•7 

184-4119 

3445-69 

202262-003 

2706-2449 

•8 

159-5932 

2580-64 

131096-512 

2026-8346 

•8 

184-7260 

3457-44 

203297-472 

2715-4733 

•9 

159-9074 

2590-81 

131872-229 

2034-8770 

•9 

185-0402 

3469-21 

204336-469 

2724-7175 

51 

160  2216 

2601 

132651 

2042-8254 

59 

185-3544 

3481 

205379 

2733-9774 

•1 

160-5357 

2611-21 

133432-831 

2050-8443 

-1 

185-6685 

3492-81 

206425-071 

2743-2529 

•2 

160-8499 

2621-44 

134217-728 

2058-8784 

•2 

185-9827 

3504-64 

207474-688 

2752-5442 

•3 

161-1640 

2631-69 

135005-697 

20669293 

•3 

186-2696 

3516-49 

208527-857 

2761-8512 

•4 

161-4782 

2641-96 

135796-744 

2074-9953 

•4 

186-6110 

3528-36 

209584-584 

2771-1739 

•5 

161-7924 

2652-25 

136590-875 

2083-0771 

•5 

186-9252 

3540-25 

210644-875 

2780-5123 

•6 

162-1065 

2662-56 

137388-096 

2091-1746 

•6 

187-2393 

3552-16 

211708-736 

2789-8664 

•7 

162-4207 

2672-89 

138188-413 

2099-2878 

•7 

187-5535 

3564-09 

212776-173 

2799-2362 

•8 

162-7348 

2683-24 

138991-832 

2107-4166 

•8 

187-8676 

3576-04 

213847-192 

2808-6218 

•9 

163-0490 

2693-61 

139798-359 

2115-5612 

•9 

188-1818 

3588-01 

214921-799 

2818-0230 

52 

163-3632 

2704 

140608 

2123-7216 

60 

188-4960 

3600 

216000 

2827-4400 

•1 

163-6773 

2714-41 

141420-761 

2131-8976 

•1 

188-8101 

3612-01 

217081-801 

2836-8720 

•2 

163-9935 

2724-84 

142236-648 

2140-0893 

•2 

189-1243 

3624-04 

218167-208 

2846-3210 

•3 

164-3056 

2735-29 

143055-667 

2148-2967 

•3 

189-4384 

3636-09 

219256-227 

2855-7850 

•4 

164-6198 

2745-76 

143877-824 

2156-5199 

•4 

189-7526 

3648-16 

220348-864 

2865-2648 

•5 

164-9340 

2756-25 

144703-125 

2164-7587 

•5 

190-0668 

3660-25 

221445-125 

2874-7603 

•6 

165-2481 

2766-76 

145531-576 

2173-0133 

•6 

190-3809 

3672-36 

222545-016 

2884-2615 

•7 

165-5623 

2777-29 

146363-183 

2181-2835 

•7 

190-6951 

3684-49 

223648-543 

2893-7984 

•8 

165-8764 

2787-84 

147197-952 

2189-5695 

•8 

191-0092 

3696-64 

224755-712 

2903-3410 

•9 

166-1906 

2798-41 

148035-889 

2197-8712 

•9 

191-3234 

3708-81 

225866-529 

2912-8993 

53 

166-5048 

2809 

148877 

2206-1886 

61 

191-6376 

3721 

226981 

2922-4734 

•1 

166-8189 

2819-61 

149721-291 

2214-5216 

•1 

191-9517 

3733-21 

228099-131 

2932-0631 

•2 

167-1331 

2830-24 

150568-768 

2222-8704 

•2 

192-2659 

3745-44 

229220-928 

2941-6685 

•3 

167-4472 

2940-89 

151419-437 

2231-2350 

•3 

192-5800 

3757-69 

230346-397 

2951-2897 

•4 

167-7614 

2851-56 

152273-304 

2239-6152 

•4 

192-8942 

3769-96 

231475-544 

2960-9265 

•5 

168-0756 

2862-25 

153130-375 

2248-0111 

•5 

193-2084 

3782-25 

232C08-375 

2970-5791 

•6 

168-3897 

2872-96 

153990-656 

2256-4227 

•6 

193-5225 

3794-56 

233744-896 

29802474 

•7 

168-7049 

2883-69 

154854-153 

2264-8701 

•7 

193-8367 

3806-89 

234885-113 

2989-9314 

•8 

169-0180 

2894-44 

155720-872 

2273-2931 

•8 

194-1508 

3819-24 

236029  032 

2999-  63CO 

•9 

169-3322 

2905-21 

156590-819 

2281-7519 

•9 

194-4650 

3831-61 

237176-659 

3009-3464 

54 

169-6464 

2916 

157464 

2290-2264 

62 

194-7792 

3844 

238328 

."019-0776 

•1 

169-9605 

2926-81 

158340-421 

2298-7165 

•1 

195-0933 

3856-41 

239483-061 

3028-8241 

•2 

170-2747 

2a37-64 

159220-088 

2307-2224 

•2 

195-4075 

3868-84 

240641-848 

3038-5809 

•3 

170-5888 

2948-49 

160103-007 

2315-7440 

•3 

195-7216 

3881-29 

241804-367 

3048-3651 

•4 

170-9030 

2959-36 

160989-184 

2324-2813 

•4 

196-0358 

3893-76 

242970-624 

3058-1591 

•5 

171-2172 

2970-25 

161878-625 

2332-8343 

•5 

196-3500 

3906-25 

244140-625 

3067-9687 

•6 

171-5313 

2981-16 

162771-336 

2341-4030 

•6 

196-6641 

3918-76 

245314-376 

3077-7941 

•7 

171-8455 

2992-09 

163667-323 

2349-9874 

•7 

196-9783 

3931-29 

246491-883 

3087-C341 

•8 

172-1596 

3003-04 

164566-592 

2358-5876 

•8 

197-2924 

3943-84 

247673-152 

3097-4919 

•9 

172-4738 

3014-01 

165469-149 

2367-2034 

•9 

197-6066 

3956-41 

248858-189 

3107-3644 

55 

172-7880 

3025 

166375 

2375-8350 

63 

197-9208 

3969 

250047 

3117-252H 

•1 

173-1021 

3036-01 

167284-151 

2384-4822 

•1 

198-2349 

3981-61 

251239-591 

3127-1564 

•2 

1734163 

3047-04 

168196-608 

2393-1452 

•2 

198-5491 

3994-24 

252435-968 

3137-0758 

•3 

173-7304 

3058-09 

169112-377 

2401-8238 

•3 

198-8632 

4006-89 

253636-137 

3147-0114 

•4 

174-0446 

3069-16 

170031-464 

2410-5182 

•4 

199-1774 

4019-56 

254840-104 

3156-9C64 

•5 

174-3588 

3080-25 

170953-875 

2419-2283 

•5 

199-4916 

4032-25 

256047-875 

3166-9291 

•6 

174-6729 

3091-36 

171879-616 

2427-9541 

•6 

199-8057 

4044-96 

257259-456 

3176-9115 

•7 

174-9771 

3102-49 

172808-693 

2436-6956 

•7 

200-1199 

4057-69 

258474-853 

3186-9097 

•8 

175-3092 

3113-64 

173741-112 

2445-4528 

•8 

200-4340 

4070-44 

259694-072 

3196-9235 

•9 

175-6154 

3124-81 

174676-879 

2454-2^57 

•9 

200-7482 

4083-21 

260917-119 

3206-9531 

56 

175-9296 

3136 

175616 

2463-0144 

64 

201-0624 

4096 

262144 

3216-9984 

•1 

176-2437 

3147-21 

176558-481 

2471-8187 

•1 

201-3765 

4108-81 

263374-721 

3227-0593 

•2 

176-5579 

3158-44 

177504-328 

2480-6387 

•2 

201-6907 

4121-64 

264609-288 

3237-13TO 

•3 

176-8720 

3169-69 

178453-547 

2489-4745 

•3 

202-0048 

4134-49 

265847-707 

3247-2284 

•4 

177-1862 

3180-96 

179406-144 

2498-3259 

•4 

202-3190 

4147-36 

267089-984 

3257-33C5 

•5 

177-5004 

3192-25 

180362-125 

2507-1931 

•5 

202-6332 

4160-25 

268336-125 

3267-4603 

•6 

177-8145 

3203-56 

181321-496 

2516-0760 

•6 

202-9473 

4173-16 

269586-136 

3277-5998 

•7 

178-1287 

3214-89 

182284-263 

2524-9736 

•7 

203-2615 

4186-09 

270840-023 

3287-7550 

•8 

178-4428 

3226-24 

183250-432 

2533-8888 

•8 

203-5756 

4199-04 

272097-792 

3297  -92t.:0 

•9 

178-7570 

3237-61 

184220-009 

2542-8188 

•9 

203-8898 

4212-01 

273359-449 

3308-1126 

CIRCLES,    ADVANCING   BY   A   TENTH. 


61 


Diara. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Square. 

Cube. 

Area. 

65 

204-2040 

4225 

274625 

3318-3150 

73 

229-3368 

5329 

389017 

4185-3966 

•1 

21)4-5181 

4238-01 

275894-451 

3328-5340 

•1 

229-6509 

5343-61 

390617-891 

4196-8712 

•2 

204-83-23 

4251-04 

277167-808 

3338-7668 

•2 

229-9651 

5358-24 

392223-168 

4208-3014 

•3 

205-1464 

4264-09 

278445-077 

3349-0162 

•3 

230-2792 

5372-89 

393832-837 

4219-8678 

•4 

205-4606 

4277-16 

279726-264 

3359-2814 

•4 

230-5934 

5387-56 

395446-904 

4231-3896 

•5 

205-7748 

4290-25 

281011-375 

3369-5623 

•5 

230*9076 

5402-25 

397065-375 

4242-9271 

•6 

206-0889 

4303-36 

282300-416 

3379-8589 

•6 

231-2217 

5416-96 

398688-256 

4254-4803 

•7 

206-4031 

4316-49 

2S3593-393 

3390-1712 

•7 

231-5359 

5431-69 

400315-553 

42G6-0493 

•8 

206-7172 

4329-64 

284890-312 

3400-4992 

•8 

231'8500 

5446-44 

401947-272 

4277-6339 

•9 

207-0314 

4342-81 

286191-179 

3410-8429 

•9 

232-1642 

5461-21 

403583-419 

42892343 

66 

207-3456 

4356 

287496 

3421-2024 

74 

232-4784 

5476 

405224 

4300-8504 

•1 

207-6597 

4369-21 

288804-781 

3431-5775 

•1 

2327925 

5490-81 

406869-021 

4^12-4821 

•2 

207-9739 

4382-44 

290117-528 

3441-9633 

•2 

233-1067 

5505-64 

408518-488 

4324-1296 

•3 

208-2880 

4395-69 

291434-247 

3452-3749 

•3 

233-4208 

5520-49 

410172-407 

4335-7928 

•4 

208-6022 

4408-96 

292754-944 

3462-7971 

•4 

233-7350 

5535-36 

411830-784 

4347-4717 

•5 

208-9164 

4422-25 

294079-625 

3473-2351 

•5 

234-0492 

5550-25 

413493-625 

4359-16C3 

•6 

209-2305 

4435-56 

295408-296 

3483-6888 

•6 

234-3633 

5565-16 

415160-936 

4370-8766 

•7 

209-5447 

4448-89 

296740-963 

3494-1640 

•7 

234-6775 

5580-09 

416832-723 

4382-6026 

•8 

209-8588 

4462-2.4 

298077-632 

3504-6432 

•8 

234-9916 

5595-04 

418508-992 

4394-3448 

•9 

210-1730 

4475-61 

299418-309 

3515-1430 

•9 

235-3058 

5610-01 

420189-749 

4406-1018 

67 

210-4872 

4489 

300763 

3525-6606 

75 

235-6200 

5625 

421875 

4417-S750 

•1 

210-8013 

4502-41 

302111-711 

3536-1928 

•1 

235-9341 

5640-01 

423564-751 

4429-6638 

•2 

211-1155 

4515-84 

303464-448 

3546-7407 

•2 

236-2483 

5655-04 

425259-008 

4441-4684 

•3 

211-4296 

4529-29 

304821-217 

3557-3043 

•3 

236-5624 

5670-09 

426957-777 

4453-2886 

•4 

211-7438 

4542-76 

306182-024 

3567-8837 

•4 

236-8766 

5685-16 

428661-064 

4465-1246 

•5 

212-0580 

4556-25 

307546-875 

3578-4787 

•5 

237-1908 

5700-25 

4303C8-875 

4476-9763 

•6 

2123721 

4569-76 

308915-776 

3589-0895 

•6 

237-5049 

5715-36 

432081-216 

4488-8437 

•7 

212-6863 

4583-29 

310288-733 

3599-7159 

•7 

237-8191 

5730-49 

433798-093 

4500-7268 

•8 

213-0004 

4596-84 

311665-752 

3610-3581 

•8 

238-1332 

5745-64 

435519-512 

4512-6256 

•9 

213-3146 

4610-41 

313046-839 

3621-0160 

•9 

238-4474 

5760-81 

437245-479 

4524-5401 

68 

213-6288 

4624 

314432 

3631-6896 

76 

238-7616 

5776 

438976 

4536-4704 

•1 

2139429 

4637-61 

315821-241 

3642-3788 

•1 

239-0757 

5791-21 

440711-081 

4548-4163 

•2 

214-2571 

4651-24 

317214-568 

36530838 

•2 

239-3899 

5806-44 

442450728 

4560-3787 

•3 

214-5712 

4664-89 

318611-987 

3663-8040 

•3 

239-7040 

5821-69 

444194-947 

4572-3553 

•4 

214-8854 

4678-56 

320013-504 

3674-5410 

•4 

240-0182 

5836-96 

445943-744 

4584-3583 

•5 

215-1996 

4692-25 

321419-125 

3685-2931 

•5 

240-3324 

5852-25 

447697-125 

459(5-3571 

•6 

215-5137 

4705-96 

322828-856 

3696-0060 

•6 

240-6465 

5867-56 

449455-096 

4(308-3816 

*7 

215-8279 

4719-69 

324242-703 

3706-8445 

•7 

240-9607 

5882-89 

451217-663 

4620-4218 

•8 

216-1420 

4733-44 

325660-672 

3717-6437 

•8 

241-2748 

5898-24 

452984-832 

4632-4776 

•9 

216-4662 

474721 

327082-769 

3728-4587 

•9 

241-5087 

5913-61 

454756-609 

4644  5492 

69 

216-7704 

4761 

328509 

3739-2894 

77 

241-9032 

5929 

456533 

4056-6366 

•1 

217-0845 

4774-81 

329939-371 

3750-1357 

•1 

242-2173 

5944-41 

458314-011 

4G68-7396 

•2 

217-3987 

4788-64 

331373-888 

3760-9978 

•2 

242-5315 

5959-84 

4C0099-648 

4680-8583 

•3 

217-7128 

4802-49 

332812-557 

3771-8756 

•3 

242-8456 

5975-29 

461889-917 

4692-9927 

•4 

218-0270 

4816-36 

334255-384 

3782-7691 

4 

243-1598 

599076 

463684-824 

4705-1429 

•5 

218-3412 

4830-25 

335702-375 

3793-6783 

•5 

243-4740 

6006-25 

465484-375 

4717-3087 

•6 

218-6553 

4844-16 

337153-536 

3804-6032 

•6 

243-7881 

6021-76 

467288-576 

4729-4903 

•7 

218-9695 

4858-09 

338608-873 

3815-5438 

•7 

244-1023 

6037-29 

469097-433 

4741-6875 

•8 

219-2836 

4872-04 

340068-392 

3826-5002 

•8 

244-4164 

6052-84 

470910-952 

4753-9605 

•9 

219-5978 

4886-01 

341532-099 

3837-4722 

•9 

244-7306 

6068-41 

472729-139 

4766-1292 

70 

219-9120 

4900 

343000 

3848-4600 

78 

245-0448 

C084 

474552 

4778-3736 

•1 

220-2261 

4914-01 

344472-101 

3859-4952 

•1 

245-3589 

6099-61 

476379-541 

4790-6336 

•2 

220-5403 

4928-04 

345948-408 

3870-4826 

•2 

245-6731 

6115-24 

478211-768 

4802-9094 

•3 

220-8544 

4942-09 

347428-927 

3881-5174 

•3 

245-9872 

6130-89 

480048-687 

4815-2010 

•4 

221-1686 

4956-16 

348913-664 

3892-5680 

•4 

246-3014 

6146-56 

481890-304 

4827-5082 

•5 

221-4828 

4970-25 

350402-625 

3903-6343 

•5 

246-6156 

6162-25 

483736-625 

4839-8311 

•G 

221-7969 

4984-36 

351895-816 

3914-7163 

•6 

246-9297 

6177-96 

485587-656 

4852-1697 

•7 

222-1111 

4998-49 

353393-243 

3925-8140 

*7 

247-2439 

6193-69 

487443-403 

4864-5241 

•8 

222-4252 

5012-64 

354894-912 

3936-9274 

•8 

247-5480 

6209-44 

489303-872 

4876-8973 

•9 

222-7394 

5026-81 

356400-829 

3948-0565 

•9 

247-8722 

6225-21 

491169-069 

4889-2799 

71 

223-0536 

5041 

357911 

3959-2014 

79 

248-1864 

6241 

493039 

4901-6814 

•1 

223-3677 

5055-21 

359425-431 

3970-3619 

•1 

248-5005 

6256-81 

494913-671 

4914-0985 

•2 

22*6819 

5069-44 

360944-128 

3981-5381 

•2 

248-8147 

6272-64 

496793-088 

4926-5314 

•3 

223-9960 

5083-69 

362467-097 

3992-7301 

•3 

249-1288 

6288-49 

498677-257 

4938-9820 

•4 

224-3102 

5097-96 

363994-344 

4003-9373 

•4 

249-4430 

6304-36 

500566-184 

4951-4443 

•5 

224-6244 

5112-25 

365525-875 

4015-1611 

•5 

249-7572 

6320-25 

502459-875 

49639243 

•6 

224-9385 

5126-56 

367061-696 

4026-4002 

•6 

250-0713 

6336-16 

504358-336 

4976-4840 

*7 

225-2527 

5140-89 

368601-813 

4037-6550 

•7 

250-3855 

6352-09 

506261-573 

4988-9314 

•8 

225-5668 

5155-24 

370146-232 

4048-9254 

•8 

250-6996 

6368-04 

508169-592 

5001-4586 

•9 

225-8810 

5169-61 

371694-959 

40'60-2116 

•9 

251-0138 

C38401 

510082-399 

5014-0014 

72 

226-1952 

5184 

373248 

4071-5136 

80 

251-3280 

6400 

512000 

502G-5COO 

•1 

226-5093 

5198-41 

374805-361 

4082-8332 

•1 

251-6421 

6416-01 

513922-401 

5039-1342 

•2 

226-8235 

5212-84 

376367-048 

4094-1645 

•2 

251-9563 

6432-04 

515849-C08 

5051-7242 

•3 

227-1376 

5227-29 

377933-007 

4105-5125 

•3 

252-2704 

6448-09 

517781-627 

5064-3298 

•4 

227-4518 

5241-76 

379503-424 

4116-8793 

•4 

252-5846 

6464-16 

519718-464 

50769552 

•5 

2-27-7660 

5256-25 

381078-125 

4128-2587 

•5 

252-8988 

6480-25 

521660-125 

5089-5883 

•6 

228-0801 

5270-76 

382657-176 

4139-6524 

•6 

253-2129 

6496-36 

523606-616 

5102-2411 

•7 

228-3943 

5285-29 

384240-583 

4151-0667 

•7 

253-5271 

6512-49 

525557-943 

5114-9096 

•8 

228-7084 

5299-84 

385828-352 

4162-4943 

•8 

253-8412 

6528-64 

527514-112 

5127-5938 

•9 

229-0226 

5314-41 

387420-489 

4173-9376 

•9 

254-1554 

6544-81 

529475-129 

5140-2937 

62 


THE   PRACTICAL   MODEL   CALCULATOR. 


fliam. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Sq^re. 

Cube. 

Area. 

81 

254-4696 

6561 

531441 

5153-0094 

89 

279-6024 

7921 

704969 

6221-1534 

•1 

254-7837 

6577-21 

533411-731 

5165-7407 

•1 

279-9165 

7938-81 

707347-971 

6235-1413 

•2 

255-0979 

6593-44 

535387-328 

5178-4877 

•2 

280-2307 

7956-64 

709732-2S8 

6249-1450 

•3 

255-4120 

6609-69 

537367-797 

5191-2505 

•3 

280-5448 

7974-49 

712121-957 

6263-1644 

•4 

255-7262 

6625-96 

539353-144 

5204-0285 

•4 

280-8590 

7992-36 

714516-984 

6277-1995 

•5 

256-0404 

6642-25 

541343-375 

5216-8231 

•5 

281-1732 

8010-25 

710917-o75 

6291-2035 

•6 

256-3545 

6658-56 

543338-496 

5229-6330 

•6 

281-4873 

8028-16 

719323-136 

6305-3168 

•7 

256-6687 

6674-89 

545338-513 

5242-4586 

•7 

281-8825 

8046-09 

721734-273 

6319-S990 

•8 

256-9828 

6691-24 

547343-432 

5255-2998 

•8 

282-1156 

8064-04 

724150-792 

63334970 

•9 

257-2970 

6707-61 

549353-259 

5268-1568 

•9 

282-4298 

8082-01 

726572-099 

6347-6813 

82 

257-6112 

6724 

551368 

5281-0296 

90 

282-7440 

8100 

729000 

6301-7400 

•1 

257-9253 

6740-41 

553387-661 

5293-9180 

•1 

283-0581 

8118-01 

731432-701 

6375-8850 

•2 

258-2395 

6756-84 

555412-248 

5306-8221 

•2 

283-3723 

8136-04 

733870-808 

6390-0458 

•3 

258-5536 

6773-29 

557441-767 

5319-7439 

•3 

283-6864 

8154-09 

736314-327 

6404-2222 

•4 

258-8646 

6789-76 

559476-224 

5332-6775 

•4 

284-0006 

8172-16 

738763-264 

6418-4144 

•5 

259  1820 

6806-25 

561515-625 

5345-6287 

•5 

584-3148 

8190-25 

741217-625 

6432-6223 

•6 

259-4961 

6822-76 

563559-976 

5358-5957 

•6 

284-6289 

8208-36 

743677-416 

6446-8474 

•7 

259-8103 

6839-29 

565609-283 

5371-5983 

•7 

284-9431 

8226-49 

746142-643 

6461-0852 

•8 

260-1244 

6855-84 

567663-552 

5384-5762 

•8 

285-2572 

8241-64 

748613-312 

6475-3402 

•9 

260-4336 

6872-41 

569722-789 

5397-5908 

•9 

285-5714 

8262-81 

751089-429 

6489-6109 

83 

260-7528 

6889 

571787 

5410-6206 

91 

285-8856 

8281 

753571 

6503-8974 

•1 

261-0669 

6905-61 

573856-191 

54236660 

•1 

286-1997 

8299-21 

756058-031 

6518-1995 

•2 

261-3811 

6922-24 

575930-368 

5436-7272 

•2 

286-5139 

8317-44 

758550-528 

6532-5173 

•3 

261-6952 

6938-89 

578009-537 

5449-8042 

•3 

286-8290 

8335-69 

761048-497 

0546-8909 

•4 

262  0094 

6955-56 

580093-704 

5462-8968 

•4 

287-1422 

8353-96 

763551-944 

6561-2081 

•5 

262-3236 

6972-25 

582182-875 

5476-0051 

•5 

287-4564 

8372-25 

766060-875 

6575-5651 

•6 

262-6376 

6988-96 

584277-056 

5489-1291 

•6 

287-7705 

8390-56 

768575-296 

6589-9458 

•7 

262-9519 

7005-69 

586376-253 

5502-2689 

•7 

288-0847 

8408-89 

771095-213 

6604-3222 

•8 

263-2840 

7022-44 

588480-472 

5515-4243 

•8 

288-3988 

8427-24 

773620-632 

6618-7542 

•9 

263-5802 

7039-21 

590589-719 

5528-5958 

•9 

288-7130 

8445-61 

776151-559 

6633-1820 

84 

263-8944 

7056 

592704 

5541-7824 

92 

289-0272 

8464 

778688 

6647-6250 

•1 

264-2085 

7072-81 

594823-321 

5554-9849 

•1 

289-3413 

8482-41 

781229-961 

6662-0848 

•2 

264-5227 

7089-64 

596947-688 

5568-2032 

•2 

289-6555 

8500-84 

783777-448 

6676-5597 

•3 

264-8368 

7106-49 

599077-107 

5581-4372 

•3 

289-9696 

8519-29 

786330-467 

6691-0161 

•4 

265-1510 

7123-36 

601211-584 

5594-6869 

•4 

290-2838 

8537-76 

788889-024 

6705-5567 

•5 

265-4652 

7140-25 

603351-125 

5607-9523 

•5 

290-5980 

8556-25 

791453-125 

6720-0787 

•6 

265-7793 

7157-16 

605495-736 

5621-2334 

•6 

290-9121 

8574-76 

794022-776 

6734-6165 

•7 

266-0935 

7174-09 

607645-423 

5634-5682 

•7 

291-2263 

8593-29 

796597-983 

6749-1699 

•8 

266-4076 

7191-04 

609800-192 

5647-8428 

•8 

291-5404 

8611-84 

799178-752 

6763-7391 

•9 

266-7218 

7208-01 

611960-049 

5661-1710 

•9 

291-8546 

8630-41 

801765-089 

6778-3240 

85 

267-0360 

7225 

614125 

5674-5150 

93 

292-1688 

8649 

804357 

1.792-9246 

•1 

267-3501 

7242-01 

616295-051 

5687-8746 

•1 

292-4829 

8667-61 

806954-491 

6807-5408 

•2 

267-6643 

7259-04 

618470-208 

5701-2500 

•2 

292-7971 

8686-24 

809557-568 

6822-1730 

•3 

267-9784 

7276-09 

620650-477 

5714-6410 

•3 

293-1112 

8704-89 

812166-237 

6836-8206 

•4 

268-2926 

7293-16 

622835-864 

5728-0478 

•4 

293-4254 

8723-56 

814780-504 

6851-4840 

•5 

268-6068 

7310-25 

625026-375 

5741-4703 

•5 

293-7396 

8742-25 

817400-375 

68C6-1631 

•6 

268-9209 

7327-36 

627222-016 

5754-9085 

•6 

294-0537 

8760-96 

820025-856 

6880-8579 

•7 

269-2351 

7344-49 

629422-793 

5768-3624 

•7 

294-3679 

8779-69 

822656-953 

C895-5685 

•8 

269-5492 

7361-64 

631628-712 

5781-8320 

•8 

294-6820 

8798-44 

825293-672 

6910-2947 

•9 

269-8634 

7378-81 

633839-779 

5795-3173 

•9 

294-9962 

8817-21 

827936-019 

6925-0367 

86 

270-1776 

7396 

636056 

5808-8184 

94 

295-3104 

8836 

830584 

6939-7944 

•1 

270-4917 

7413-21 

638277-381 

5822-3351 

•1 

295-6245 

8854-81 

833237-621 

6954-5677 

•2 

270-8059 

7430-44 

640503-928 

5835-8675 

•2 

295-9387 

8873-64 

835896-888 

6969-3568 

•3 

271-1200 

7447-69 

642735-647 

5849-4157 

•3 

296-2436 

8892-49 

838561-807 

6984-1614 

•4 

271-4342 

7464-96 

644972-544 

5862-9795 

•4 

296-5670 

8911-36 

841232-384 

6998-9821 

•5 

271-7484 

7482-25 

647214-625 

5876-5591 

•5 

296-8812 

8930-25 

843908-625 

7013-8183 

•6 

272-0665 

7499-56 

649461-896 

5890-1541 

•6 

297-1953 

8949-16 

846590-536 

7028-6702 

•7 

272-3767 

7516-89 

651714-363 

5903-7654 

•7 

297-5095 

8968-09 

849278-123 

7043-5025 

•8 

272-6908 

7534-24 

653972-032 

5917-3920 

•8 

297-8236 

8987-04 

851971-392 

7058-4180 

•9 

273-0050 

7551-61 

656234-909 

5931-0344 

•9 

298-1378 

9006-01 

854670-349 

7073-3202 

87 

273-3192 

7569 

658503 

5944-6926 

95 

298-4520 

9025 

857375 

7088-2350 

•1 

273-6333 

7586-41 

660776-311 

5958-3644 

•1 

298-7661 

9044-01 

860085-351 

7103-1654 

•2 

273-9875 

7603-84 

663054-848 

5972-0559 

•2 

299-0723 

9063-04 

862801-408 

7118-1116 

•3 

274-2616 

7621-29 

665338-617 

5985-7691 

•3 

299-3944 

9082-09 

865523-177 

7133-0734 

•4 

274-5758 

7638-76 

667627-624 

5999-4821 

•4 

299-7086 

9101-16 

868250-664 

7148-0510 

•5 

274-8900 

7656-25 

669921-875 

6013-2187 

•5 

300-0228 

9120-25 

870983-875 

7163-0443 

•6 

275-2041 

7673-76 

672221-376 

6026-9711 

•6 

300-3369 

9139-36 

873722-816 

7178-0533 

•7 

275-5183 

7691-29 

674526-133 

6040-7391 

•7 

300-6511 

9158-49 

876467-493 

7193-0780 

•8 

275-8324 

7708-84 

676836-152 

6054-5149 

•8 

300-9652 

9177-64 

879217-912 

7208-1184 

•9 

276-1466 

7726-41 

679151-439 

6068-3224 

•9 

301-2794 

9196-81 

881974-079 

7223-1745 

88 

276-4608 

7744 

681472 

6082-1376 

96 

301-5936 

9216 

884736 

7238-2464 

•1 

276-7749 

7761-61 

683797-841 

6095-9684 

•1 

301-9077 

9235-21 

887503-681 

7253-3339 

•2 

277-0891 

7779-24 

686128-908 

6109-8150 

•2 

302-2219 

9254-44 

890277-128 

7268-4371 

•3 

277-4032 

7796-89 

6S8465-387 

6123-6774 

•3 

302-5360 

9273-69 

893056-347 

7283-5561 

•4 

277-7174 

7814-56 

690807-104 

6137-5554 

•4 

302-8502 

9292-96 

895841-344 

7298-C907 

•5 

278-0316 

7832-25 

693154-122 

6151-4491 

•5 

303-1644 

9312-25 

898632-125 

7313-8411 

•6 

278-3457 

7849-96 

695506-456 

6165-3585 

•6 

303-4785 

9331-56 

901428-696 

7329-0072 

•7 

278-6599 

7867-69 

697864-103 

6179-2837 

•7 

303-7927 

935089 

904231-063 

7344-1890 

•8 

278-9750 

7885-44 

700227-072 

6193-2215 

•8 

304-1068 

9370-24 

907039-232 

7359-3864 

•9 

279-2882 

7903-21 

702595-369 

6207-1811 

•9 

304-4210 

9389-61 

909853-209 

7374-5996 

TABLE   OF   THE  LENGTH   OF   CIRC1 


TTNIVF/RSTTS 

• 


Diam. 

Circum. 

Square. 

Cube. 

Area. 

Diam. 

Circum. 

Square. 

Cube. 

Area. 

97 

304-7352 

9409 

912673 

7389-8286 

•6 

309-7617 

9721-96 

958585-256 

7635-6273 

•1 

305-0493 

9428-41 

915498-611 

7405-0732 

•7 

310-0759 

9741-69 

961504-803 

7651-1933 

•2 

305-3635 

9447-84 

918330-048 

7420-3335 

•8    310-3960 

9761-44 

964430-272 

7666-6349 

•3 

305-6776 

9467-29 

921167-317 

7435-6095 

•9 

310-7042 

9781-21 

967361-669 

7682-1623 

•4 

305-9918 

9486-76 

924010-424 

7450-9013 

99 

311-0184 

9801 

970299 

7697-7054 

•5 

306-3060 

9506-25 

926859-375 

7466-2087 

•1 

311-3325 

9820-81 

973242-271 

7713-2641 

•6 

306-6201 

9525-76 

929714-176 

7481-5319 

•2 

311-6467 

9840-64 

976191-488 

7728-8386 

•7 

306-9363 

9545-29 

932574-833 

7496-8707 

•3 

311-9008 

9860-49 

979146-657 

7744-4288 

•8 

307-2484 

9564-84 

935441-352 

7512-2253 

•4 

312-2750 

9880-36 

982107-784 

7760-0347 

•9 

307-5626 

9584-41 

938313-739 

7527-5956 

•5 

812-5892 

9900-25 

985074-875 

7775-6563 

98 

307-8768 

9604 

941192 

7542-9816 

•6 

312-9033 

9920-16 

988047-936 

7791-2936 

•1 

308-1909 

9623-61 

944076-141 

7558-3832 

•7 

313-2175 

9940-09 

991026-973 

7806-9466 

•2 

308-5051 

9643-24 

946966-168 

7573-8006 

•8 

313-5116 

9960-04 

994011-992 

7822-6154 

•3 

308-8192 

9662-89 

949862-087 

7589-2338 

•9 

313-8458 

9980-01 

997002-999 

7838-2998 

•4 

309-1334 

9682-56 

952763-904 

7604-6826 

100 

314-1600 

10000 

1000000 

7854-0000 

•5 

309-4476 

9702-25 

955671-625 

7620-1471 

A  TABLE  of  the  Length  of  Circular  Arcs,  radius  being  unity. 


Degree. 

Length. 

Degree. 

Length. 

Min. 

Length. 

Sec. 

Length. 

1 

0-0174583 

~60~ 

1-0471976 

1 

0-0002909 

1 

0-000048 

2 

0-0349066 

70 

1-2217305 

2 

0-0005818 

2 

0-000097 

3 

0-0523599 

80 

1-3962634 

3 

0-0008727 

3 

0-0000145 

4 

0-0698132 

90 

1-5707963 

4 

0-0011636 

4 

0-0000194 

5 

0-0872665 

100 

1-7453293 

5 

0-0014544 

5 

0-0000242 

6 

0-1047198 

120 

2-0943951 

6 

0-0017453 

6 

0-0000291 

7 

0-1221730 

150 

2-6179939 

7 

0-0020362 

7 

0-0000339 

8 

0-1396263 

180 

3-1415927 

8 

0-0023271 

8 

0-0000388 

9 

0-1570796 

210 

3-6651914 

9 

0-0026180 

9 

0-0000436 

10 

0-1745329 

240 

4-1887902 

10 

0-0029089 

10 

0-0000485 

20 

0-3490659 

270 

4-7123890 

20 

0-0058178 

20 

0-0000970 

30 

0-5235988 

300 

6-2359878 

30 

0-0087266 

30 

0-0001454 

40 

0-6981317 

330 

5-7595865 

40 

0-0116355 

40 

0-0001939 

50 

0-8726646 

360 

6-2831853 

50 

00145444 

50 

0-0002424 

42/58//? 


Required  the  length  of  a  circular  arc  of  37 ( 

30°  =  0-5235988 

7°  =  0-1221730 

40'  =  0-0116355 

2'  =  0-0020368 

50"  =  0-0002424 

8"  =  0-0000388 

The  length  0-6582703" required  in  terms  of  the 
radius. 

1207°  Fahrenheit  =  1°  of  Wedgewood's  pyrometer.  Iron  melts 
at  about  166°  Wedgewood ;  200362°  Fahrenheit. 

Sound  passes  in  air  at  a  velocity  of  1142  feet  a  second,  and  in 
water  at  a  velocity  of  4700  feet. 

Freezing  water  gives  out  140°  of  heat,  and  may  be  cooled  as 
low  as  20°.  All  solids  absorb  heat  when  becoming  a  fluid,  and  the 
quantity  of  heat  that  renders  a  substance  fluid  is  termed  its  caloric 
of  fluidity,  or  latent  heat.  Fluids  in  vacuo  boil  with  124°  less 
heat,  than  when  under  the  pressure  of  the  atmosphere. 


64 


THE   PRACTICAL   MODEL    CALCULATOR. 


AREAS  of  the  Segments  and  Zones  of  a  Circle  of  which  the  DIAMETER 
is  Unity,  and  supposed  to  be  divided  into  1000  equal  parts. 


Height. 

Area  of 
Segment. 

Area  of 
Zone. 

Height. 

Area  of 
Segment. 

Area  of 
Zone. 

Height. 

Area  of 
Segment. 

Area  of 
Zone. 

•001 

•000042 

•001000 

•051 

•015119 

•050912 

•101 

•041476 

•100309 

•002 

•000119 

•002000 

•052 

•015561 

•051906 

•102 

•042080 

•101288 

•003 

•000219 

•003000 

•053 

•016007 

•052901 

•103 

•052687 

•102267 

•004 

•000337 

•004000 

•054 

•016457 

•053895 

•104 

•043296 

•103246 

•005 

•000470 

•005000 

•055 

•016911 

•054890 

•105 

•043908 

•104223 

•006 

•000618 

•006000 

•056 

•017369 

•055883 

•106 

•044522 

•105201 

•007 

•000779 

•007000 

•057 

•017831 

•056877 

•107 

•045139 

•106178 

•008 

•000951 

•008000 

•058 

•018296 

•057870 

•108 

•045759 

•10715o 

•009 

•001135 

•009000 

•059 

•018766 

•058863 

•109 

•046381 

•108131 

•010 

•001329 

•010000 

•060 

•019239 

•059856 

•110 

•047005 

•109107 

•on 

•001533 

•011000 

•061 

•019716 

•060849 

•111 

•047632 

•110082 

•012 

•001746 

•011999 

•062 

•020196 

•061841 

•112 

•048262 

•111057 

•013 

•001968 

•012999 

•063 

•020680 

•062833 

•113 

•048894 

•112031 

•014 

•002199 

•013998 

•064 

•021168 

•063825 

•114 

•049528 

•113004 

•015 

•002438 

•014998 

•065 

•021659 

•064817 

•115 

•050165 

•113978 

•016 

•002685 

•015997 

•066 

•022154 

•065807 

•116 

•050804 

•114951 

•017 

•002940 

•016997 

•067 

•022652 

•066799 

•117 

•051446 

•115924 

•018 

•003202 

•017996 

•068 

•023154 

•067790 

•118 

•052090 

•116896 

•019 

•003471 

•018996 

•069 

•023659 

•068782 

•119 

•052736 

•117867 

•020 

•003748 

•019995 

•070 

•024168 

•069771 

•120 

•053385 

•118838 

•021 

•004031 

•020994 

•071 

•024680 

•070761 

•121 

•054036 

•119809 

•022 

•004322 

•021993 

•072 

•025195 

•071751 

•122 

•054689 

•120779 

•023 

•004618 

•022992 

•073 

•025714 

•072740 

•123 

•055345 

•121748 

•024 

•004921 

023991 

•074 

•026236 

•073729 

•124 

•056003 

•122717 

•025 

•005230 

•024990 

•075 

•026761 

•074718 

•125 

•056663 

•123686 

•026 

•005546 

•025989 

•076 

•027289 

•075707 

•126 

•057326 

•124654 

•027 

•005867 

•026987 

•077 

•027821 

•076695 

•127 

•057991 

•125621 

•028 

•006194 

•027986 

•078 

•028356 

•077683 

•128 

•058658 

•126588 

•029 

•006527 

•028984 

•079 

•028894 

•078670 

•129 

•059327 

•127555 

•030 

•006865 

•029982 

•080 

•029435 

•079658 

•130 

•059999 

•128521 

•031 

•007209 

•030980 

•081 

•029979 

•080645 

•131 

•060672 

•129486 

•032 

•007558 

•031978 

•082 

•030526 

•081631 

•132 

•061348 

•130451 

•033 

•007913 

•032976 

•083 

•031076 

•082618 

•133 

•062026 

•131415 

•034 

•008273 

•033974 

•084 

•031629 

•083604 

•134 

•062707 

•132379 

•035 

•008638 

•034972 

•085 

•032186 

•084589 

•135 

•063389 

•133342 

•036 

•009008 

•035969 

•086 

•032745 

•085574 

•136 

•064074 

•134304 

•037 

•009383 

•036967 

•087 

•033307 

•086559 

•137 

•064760 

•135266 

•038 

•009763 

•037965 

•088 

•033872 

•087544 

•138 

•065449 

•136228 

•039 

•010148 

•038962 

•089 

•034441 

•088528 

•139 

•066140 

•137189 

•040 

•010537 

•039958 

•090 

•035011 

•089512 

•140 

•066833 

•138149 

•041 

•010931 

•040954 

•091 

•035585 

•090496 

•141 

•067528 

•139109 

•042 

•011330 

•041951 

•092 

•036162 

•091479 

•142 

•068225 

•140068 

•043 

•011734 

•042947 

•093 

•036741 

•092461 

•143 

•068924 

•141026 

•044 

•012142 

•043944 

•094 

•037323 

•093444 

•144 

•069625 

•141984 

•045 

•012554 

•044940 

•095 

•037909 

•094426 

•145 

•070328 

•142942 

•046 

•012971 

•045935 

•096 

•038496 

•095407 

•146 

•071033 

•143898 

•047 

•013392 

•046931 

•097 

.039087 

•096388 

•147 

•071741 

•144854 

•048 

•013818 

•047927 

•098 

.039680 

•097369 

•148 

•072450 

•145810 

•049 

•014247 

•048922 

•099 

.040276 

•098350 

•149 

•073161 

•146765 

•050 

•014681 

•049917 

•100 

.040875 

•099330 

•150 

•073874 

•147719 

AREAS    OF   THE    SEGMENTS   AND    ZONES    OF   A   CIRCLE. 


65 


Height. 

Area  of  Seg. 

Area  of  Zone. 

Height. 

Area  of  Seg. 

Area  of  Zone. 

Height. 

Area  of  Seg. 

Area  of  Zone. 

•151 

•074589 

•148674 

•206 

•116650 

•200915 

•261 

•103140 

•248608 

•152 

•075306 

•149625 

•207 

•117460 

•200924 

•262 

•164019 

•249461 

•153 

•076026 

•150578 

•208 

•118271 

•201835 

•263 

•164899 

•250212 

•154 

•076747 

•151530 

•209 

•119083 

•202744 

•264 

•165780 

•251162 

•155 

•077469 

•152481 

•210 

•119897 

•203652 

•265 

•166663 

•252011 

•156 

•078194 

•153431 

•211 

•120712 

•204559 

•266 

•167546 

•252851 

•157 

•078921 

•154381 

•212 

•121529 

•205465 

•267 

•168430 

•253704 

•158 

•079649 

•155330 

•213 

•122347 

•206370 

•268 

•169315 

•254549 

•159 

•080380 

•156278 

•214 

•123167 

•207274 

•269 

•170202 

•255392 

•160 

•081112 

•157226 

•215 

•123988 

•208178 

•270 

•171080 

•256235 

•161 

•081846 

•158173 

•216 

•124810 

•209080 

•271 

•171978 

•257075 

•162 

•082582 

•159119 

•217 

'125634 

•209981 

•272 

•172867 

•257915 

•163 

•083320 

•160065 

•218 

•126459 

•210882 

•273 

•173758 

•258754 

•164 

•084059 

•161010 

•219 

•127285 

•211782 

•274 

•174649 

•259591 

•165 

•084801 

•161954 

•220 

•128113 

•212680 

•275 

•175542 

•260427 

•166 

•085544 

•162898 

•221 

•128942 

•213577 

•276 

•176435 

•261261 

•167 

•086289 

•163841 

•222 

•129773 

•214474 

•277 

•177330 

•262094 

•168 

•087036 

•165784 

•223 

•130605 

•215369 

•278 

•178225 

•262926 

•169 

•087785 

•165725 

•224 

•131438 

•216264 

•279 

•179122 

•263757 

•170 

•088535 

•166666 

•225 

•132272 

•217157 

•280 

•180019 

•264586 

•171 

•089287 

•167606 

•226 

•133108 

•218050 

•281 

•180918 

•265414 

•172 

•090041 

•168549 

•227 

•133945 

•218941 

•282 

•181817 

•266240 

•173 

•090797 

•160484 

•228 

•134784 

•219832 

•283 

•182718 

•267065 

•174 

•091554 

•170422 

•229 

•135624 

•220721 

•284 

•183619 

•267889 

•175 

•092313 

•171359 

•230 

•136465 

•221610 

•285 

•184521 

•268711 

•176 

•093074 

•172295 

•231 

•137307 

•222497 

•286 

•185425 

•269532 

•177 

•093836 

•173231 

•232 

•138150 

•223354 

•287 

•186329 

•270352 

•178 

•094601 

•174166 

•233 

•138995 

•224269 

•288 

•187234 

•271170 

•179 

•095366 

•175100 

•234 

•139841 

•225153 

•289 

•188140 

•271987 

•180 

•096134 

•176033 

•235 

•140688 

•226036 

•290 

•189047 

•272802 

•181 

•096903 

•176966 

•236 

•141537 

•226919 

•291 

•189955 

•273616 

•182 

•097674 

•177897 

•237 

•142387 

•227800 

•292 

•190864 

•274428 

•183 

•098447 

•178828 

•238 

•143238 

•228680 

•293 

•191775 

•275239 

•184 

•099221 

•179759 

•239 

•144091 

•229559 

•294 

•192684 

•276049 

•185 

•099997 

•180688 

•240 

•144944 

•230439 

•295 

•193596 

•276857 

•186 

•100774 

•181617 

•241 

•145799 

•231313 

•296 

•194509 

•277664 

•187 

•101553 

•182545 

•242 

•146655 

•232189 

•297 

•195422 

•278469 

•188 

•102334 

•183472 

•243 

•147512 

•233063 

•298 

•196337 

•279273 

•189 

•103116 

•184398 

•244 

•148371 

•233937 

•299 

•197252 

•280075 

•190 

•103900 

•185323 

•245 

•149230 

•234809 

•300 

•198168 

•280876 

•191 

•104685 

•186248 

•246 

•150091 

•235680 

•301 

•199085 

•281675 

•192 

•105472 

•187172 

•257 

•150953 

•236550 

•302 

•200003 

•282473 

•193 

•106261 

•188094 

•248 

•151816 

•237419 

•303 

•200922 

•283269 

•194 

•107051 

•189016 

•249 

•152680 

•238287 

•304 

•201841 

•284063 

•195 

•107842 

•189938 

•250 

•153546 

•239153 

•305 

•202761 

•284857 

•196 

•108636 

•190858 

•251 

•154412 

•240019 

•306 

•203683 

•285648 

•197 

•109430 

•191777 

•252 

•155280 

•240883 

•307 

•204605 

•286438 

•198 

•110226 

•192696 

•253 

•156149 

•241746 

•308 

•205527 

•287227 

•199 

•111024 

•193614 

•254 

•157019 

•242608 

•309 

•206451 

•288014 

•200 

•111823 

•194531 

•255 

•157890 

•243469 

•310 

•207376 

•288799 

•201 

•112624 

•195447 

•256 

•158762 

•244328 

•311 

•208301 

•289583 

•202 

•113426 

•196362 

•257 

•159636 

•245187 

•312 

•209227 

•290365 

•203 

•114230 

•197277 

•258 

•160510 

•246044 

•313 

•210154 

•291146 

•204 

•115035 

•198190 

•259 

•161386 

•246900 

•314 

•211082 

•291925 

•205 

•115842 

•199103 

•260 

•162263 

•247755 

•315 

•212011 

•292702 

I 

THE   PRACTICAL  MODEL  CALCULATOR. 


Height. 

Area  of  Seg. 

Area  of  Zone. 

Height. 

Area  of  Seg.|AreaofZone 

Height.  Area  of  Seg. 

Area  of  Zone. 

•316 

•212940 

•293478 

•371 

•265144 

•333372 

•426 

•318970 

•366463 

•317 

•213871 

•294252 

•372 

•266111 

•334041 

•427 

•319959 

•366985 

•318 

•214802 

•295025 

-373 

•267078 

•334708 

•428 

•320948 

•367504 

•319 

•215733 

•295796 

•374 

•268045 

•335373 

•429 

•321938 

•368019 

•320 

•216666 

•296565 

•875 

•269013 

•336036 

•430 

•322928 

•368531 

•321 

•217599 

•297333 

•376 

•269982 

•336696 

•431 

•323918 

•369040 

•322 

•218533 

•298098 

•377 

•270951 

•337354 

•432 

•324909 

•369545 

•323 

•219468 

•298863 

•378 

•271920 

•338010  ' 

•433 

•325900 

•370047 

•324 

•220404 

•299625 

•379 

•272890 

•338663 

•434 

•326892 

•370545 

-325 

•221340 

•300386 

•380 

•273861 

•339314 

•435 

•327882 

•371040 

-326 

•222277 

•301145 

•381 

•274832 

•339963 

•436 

•328874 

•371531 

-327 

•223215 

•301902 

•382 

•275803 

•340609 

•437 

•329866 

•372019 

-328 

-224154 

•302658 

•383 

•276775 

•341253 

•438 

•330858 

•372503 

-329 

-225093 

•303412 

•384 

•277748 

•341895 

•439 

•331850 

•372983 

•330 

•226033 

•304164 

•385 

•278721 

•342534 

•440 

•332843 

•373460 

•331 

-226974 

•304914 

•886 

•279694 

•343171 

•441 

•333836 

•373933 

•332 

•227915 

•305663 

•387 

•280668 

•343805 

•442 

•334829 

•374403 

•333 

•228858 

•306410 

•388 

•281642 

•344437 

•443 

•335822 

•374868 

•334 

•229801 

•307155 

•389 

•282617 

•345067 

•444 

•336816 

•375330 

•335 

•230745 

•307898 

•890 

-283592 

•345694 

•445 

•337810 

•375788 

•336 

•231689 

-308640 

•391 

•284568 

•346318 

•446 

•338804 

•376242 

•337 

•232634 

•309379 

•392 

•285544 

•346940 

•447 

•339798 

•376692 

•338 

•233580 

•310117 

•393 

•286521 

•347560 

•448 

•340793 

•377138 

•339 

•234526 

•310853 

•394 

•287498 

•348177 

•449 

•341787 

•377580 

•340 

•235473 

•311588 

-395 

•288476 

•348791 

•450 

•342782 

•378018 

•341 

•236421 

-312319 

•396 

•289453 

•349403 

•451 

•343777 

•378452 

•342 

•237369 

•313050 

•397 

•290432 

•350012 

•452 

•344772 

•378881 

•343 

•238318 

•313778 

•398 

•291411 

•350619 

•453 

•345768 

•379307 

•344 

•239268 

•314505 

•399 

•292390 

•351223 

•454 

•346764 

•379728 

•345 

•240218 

•315230 

•400 

•293369 

•351824 

•455 

•347759 

•380145 

•346 

•241169 

-315952 

-401 

•294349 

•352423 

•456 

•348755 

•380557 

•347 

•242121 

•316673 

•402 

•295330 

•353019 

•457 

•349752 

•380965 

•348 

•243074 

•317393 

-403 

•296311 

•353612 

•458 

•350748 

•381369 

•349 

•244026 

•318110 

•404 

•297292 

•354202 

•459 

•351745 

•381768 

•350 

•244980 

•318825 

•405 

•298273 

•354790 

•460 

•352742 

•382162 

•351 

•245934 

•319538 

•408 

•299255 

•355376 

•461 

•353739 

•382551 

•352 

•246889 

•320249 

•407 

•300238 

•355958 

•462 

•354736 

•382936 

•353 

•247845 

•320958 

•408 

•301220 

•356537 

•463 

•355732 

•383316 

•354 

•248801 

•321666 

•409 

•302203 

•357114 

•464 

•356730 

•383691 

•355 

•249757 

•322371 

•410 

•303187 

•357688 

•465 

•357727 

•384061 

•356 

•250715 

•323075 

•411 

•304171 

•358258 

•466 

•358725 

•384426 

•357 

•251673 

•323775 

•412 

•305155 

•358827 

•467 

•359723 

•384786 

•358 

•252631 

•324474 

•413 

•306140 

•359392 

•468 

•360721 

•385144 

•359 

•253590 

•325171 

•414 

•307125 

•359954 

•469 

•361719 

•385490 

•360 

•254550 

•325866 

•415 

•308110 

•360513 

•470 

•362717 

•385834 

•381 

•255510 

•326559 

•416 

•309095 

•361070 

•471 

•363715 

•386172 

•362 

•256471 

•327250 

•417 

•310081 

•361623 

•472 

•364713 

•386505 

•363 

•257433 

•327939 

•418 

•311068 

•362173 

•473 

•365712 

•386832 

•364 

•258395 

•328625 

•419 

•312054 

•362720 

•474 

•366710 

•387153 

•365 

•259357 

•329310 

•420 

•313041 

•363264 

•475 

•367709 

•387469 

•366 

•260320 

•329992 

•421 

•314029 

•363805 

•476 

•368708 

•387778 

•367 

•261284 

•330673 

•422 

•315016 

•364343 

•477 

•369707 

•388081 

•368 

•262248 

•331351 

•423 

•316004 

•364878 

•478 

•370706 

•388377 

•369 

•263213 

•332027 

•424 

•316992 

•365410 

•479 

•371704 

•388669 

•370 

•264178 

•332700 

•425 

•317981 

•365939 

•480 

•372704 

•388951 

RULES   FOR  FINDING  THE  AREA  OF  A  CIRCULAR  ZONE,  ETC.      67 


Height. 

Area  of  Seg 

Area  of  Zone. 

Height. 

Area  of  Seg. 

Area  of  Zone. 

Height. 

Area  of  Seg. 

Area  of  Zone. 

•481 

•482 
•483 
•484 
•435 

•486 
•487 
•488 
•489 
•490 

•373703 
•374702 
•375702 
•376702 
•377701 

•378701 
•379700 
•380700 
•381699 
•382699 

•389228 
•389497 
•389759 
•390014 
•390261 

•390500 
•390730 
•390953 
•391166 
•391370 

•491 

•492 
•493 
•494 
•495 

•383699 
•384699 
•385699 
•386699 
•387699 

•391564 
•391748 
•391920 
•392081 
•392229 

•496 
•497 
•498 
•499 
•500 

•388699 
•389699 
•390699 
•391699 
•392699 

•392362 
•392480 
•392580 
•392657 
•392699 

To  find  the  'area  of  a  segment  of  a  circle. 
RULE.  —  Divide  the  height,  or  versed  sine, 
by  the  diameter  of  the  circle,  and  find  the 

mint.iftnt1.  in  t,hf»  nnlnmn  of  Vim  or  fits 

Then  take  out  the  corresponding  area,  in  the  column  of  areas, 
and  multiply  it  by  the  square  of  the  diameter ;  this  will  give  the 
area  of  the  segment. 

Required  the  area  of  a  segment  of  a  circle,  whose  height  is  3J- 
feet,  and  the  diameter  of  the  circle  50  feet. 

31  =  3-25 ;   and  3-25  -H  50  =  -065. 

•065,  by  the  Table,  =  -021659 ;  and  -021659  x  502  =  54-147500, 
the  area  required. 

To  find  the  area  of  a  circular  zone. 

RULE  1. — When  the  zone  is  less  than  a  semi-circle,  divide  the 
height  by  the  longest  chord,  and  seek  the  quotient  in  the  column 
of  heights.  Take  out  the  corresponding  area,  in  the  next  column 
on  the  right  hand,  and  multiply  it  by  the  square  of  the  longest  chord. 

Required  the  area  of  a  zone  whose  longest  chord  is  50,  and  height  15. 
15  -j-  50  =  -300 ;  and  -300,  by  the  Table,  =  -280876. 
Hence  -280876  X  502  =  702-19,  the  area  of  the  zone. 

RULE  2. — When  the  zone  is  greater  than  a  semi-circle,  take  the 
height  on  each  side  of  the  diameter  of  the  circle. 

Required  the  area  of  a  zone,  the  diameter  of  the  circle  being  50, 
and  the  height  of  the  zone  on  each  side  of  the  line  which  passes 
through  the  diameter  of  the  circle  20  and  15  respectively. 

20  -r-  50  =  -400 ;  -400,  by  the  Table,  =  -351824 ;  and  -351824  x 
502  =  879-56. 

15  --  50  =  -300 ;  -300,  by  the  Table,  =-280876  ;  and  -280876  X 
502  =  702-19.  Hence  879-56  +  702-19  =  1581-75. 

Approximating  rule  to  find  the  area  of  a  segment  of  a  circle. 

RULE. — Multiply  the  chord  of  the  segment  by  the  versed  sine, 
divide  the  product  by  3,  and  multiply  the  remainder  by  2. 

Cube  the  height,  or  versed  sine,  find  how  often  twice  the  length 
of  the  chord  is  contained  in  it,  and  add  the  quotient  to  the  former 
product ;  this  will  give  the  area  of  the  segment  very  nearly. 

Required  the  area  of  the  segment  of  a  circle,  the  chord  being  12, 
and  the  versed  sine  2. 

12  x  2  =  24 ;  y  =  8 ;  and  8  X  2  =  16. 

23  +  24  =  -3333. 
Hence  16  +  -3333=16-3333,  the  area  of  the  segment  very  nearly. 


68 


PROPORTIONS   OF   THE    LENGTHS   OF   CIRCULAR   ARCS, 


Height 
of 

Length 
of 

Height 

Length 

Height 

Length 
of 

Height 
ot 

Length 
of 

Height 
of 

Length 
Arc. 

•100 

1-02645 

•181 

1-08519 

•261 

1-17275 

•341 

1-28583 

•421 

1-42041 

•101 

1-02098 

•182 

1-08611 

•262 

1-17401 

•342 

1-28739 

•422 

1-42222 

•102 

1-02752 

•183 

1-08704 

•263 

1-17527 

•343 

1-28895 

•423 

1-42402 

•103 

1-02806 

•184 

1-08797 

•264 

1-17655 

•344 

1-29052 

•424 

1-42583 

•104 

1-02860 

•185 

1-08890 

•265 

1-17784 

•345 

1-29209 

•425 

1-42764 

•105 

1-02914 

•186 

1-08984 

•266 

1-17912 

•346 

1-29366 

•426 

1-42945 

•106 

1-02970 

•187 

1-09079 

•267 

1-18040 

•347 

1-29523 

•427 

1-43127 

•107 

1-03026 

•188 

1-09174 

•268 

1-18162 

•348 

1-29681 

•428 

1-43309 

•108 

1-03082 

•189 

1-09269 

•269 

1-18294 

•349 

1-29839 

•429 

1-43491 

•109 

1-03139 

•190 

1-09365 

••270 

1-18428 

•350 

1-29997 

•430 

1-43673 

•110 

1-03196 

•191 

1-09461 

•271 

1-18557 

•351 

1-30156 

•431 

1-43856 

•111 

1-03254 

•192 

1-09557 

•272 

1-18688 

•352 

1-30315 

•432 

1-44039 

•112 

1-03312 

•193 

1-09654 

•273 

1-18819 

•353 

1-30474 

•433 

1-44222 

•113 

1-03371 

•194 

1-09752 

•274 

1-18969 

•354 

1-30634 

•434 

1-44405 

•114 

1-03430 

•195 

1-09850 

•275 

1-19082 

•355 

1-30794 

•435 

1-44589 

•115 

1-03490 

•196 

1-09949 

•276 

1-19214 

•356 

1-30954 

•436 

1-44773 

•116 

1-03551 

•197 

1-10048 

•277 

1-19345 

•357 

1-31115 

•437 

1-44957 

•117 

1-03611 

•198 

1-10147 

•278 

1-19477 

•358 

1-31276 

•438 

1-45142 

•118 

1-03672 

•199 

1-10247 

•279 

1-19610 

•359 

1-31437 

•439 

1-45327 

•119 

1-03734 

•200 

1-10348 

•280 

1-19743 

•360 

1-31599 

•440 

1-45512 

•120 

1-03797 

•201 

1-10447 

•281 

1-19887 

•361 

1-31761 

•441 

1-45697 

•121 

1-03860 

•202 

1-10548 

•282 

1-20011 

•362 

1-31923 

•442 

1-45883 

•122 

1-03923 

•203 

1-10650 

•283 

1-20146 

•363 

1-32086 

•443 

1-46069 

•123 

1-03987 

•204 

1-10752 

•284 

1-20282 

•364 

1-32249 

•444 

1-46255 

•124 

1-04051 

•205 

1-10855 

•285 

1-20419 

•365 

1-32413 

•445 

1-46441 

•125 

1-04116 

•206 

1-10958 

•286 

1-20558 

•366 

1-32577 

•446 

1-46628 

•126 

1-04181 

•207 

1-11062 

•287 

1-20696 

•367 

1-32741 

•447 

1-46815 

•127 

1-04247 

•208 

1-11165 

•288 

1-20828 

•368 

1-32905 

•448 

1-47002 

•128 

1-04313 

•209 

1-11269 

•289 

1-20967 

•369 

1-33069 

•449 

1-47189 

•129 

1-04380 

•210 

1-11374 

•290 

1-21202 

•370 

1-33234 

•450 

1-47377 

•130 

1-04447 

•211 

1-11479 

•291 

1-21239 

•371 

1-33399 

•451 

1-47565 

•131 

1-04515 

•212 

1-11584 

•292 

1-21381 

•372 

1-33564 

•452 

1-47753 

•132 

1-04584 

•213 

1-11692 

•293 

1-21520 

•373 

1-33730 

•453 

1-47942 

•133 

1-04652 

•214 

1-11796 

•294 

1-21658 

•374 

1-33896 

•454 

1-48131 

•134 

1-04722 

•215 

1-11904 

•295 

1-21794 

•375 

1-34063 

•455 

1-48320 

•135 

1-04792 

•216 

1-12011 

•296 

1-21926 

•376 

1-34229 

•456 

1-48509 

•136 

1-04862 

•217 

1-12118 

•297 

1-22061 

•377 

1-34396 

•457 

1-48699 

•137 

1-04932 

•218 

1-12225 

•298 

1-22203 

•378 

1-34563 

•458 

1-48889 

•138 

1-05003 

•219 

1-12334 

•299 

1-22347 

•379 

1-34731 

•459 

1-49079 

•139 

1-05075 

•220 

1-12445 

•300 

1-22495 

•380 

1-34899 

•460 

1-49269 

•140 

1-05147 

•221 

1-12556 

•301 

1-22635 

•381 

1-35068 

•461 

1-49460 

•141 

1-05220 

•222 

1-12663 

•302 

1-22776 

•382 

1-35237 

•462 

1-49661 

•142 

1-05293 

•223 

1-12774 

•303 

1-22918 

•383 

1-35406 

•463 

1-49842 

•143 

1-05367 

•224 

1-12885 

•304 

1-23061 

•384 

1-35575 

•464 

1-50033 

•144 

1-05441 

•225 

1-12997 

•305 

1-23205 

•385 

1-35744 

•465 

1-50224 

•145 

1-05516 

•226 

1-13108 

•306 

1-23349 

•386 

1-35914 

•466 

1-50416 

•146 

1-05591 

•227 

113219 

•307 

1-23494 

•387 

1-36084 

•467 

1-50608 

•147 

1-05667 

•228 

1-13331 

•308 

1-23636 

•388 

1-36254 

•468 

1-50800 

•148 

1-05743 

•229 

1-13444 

•309 

1-23780 

•389 

1-36425 

•469 

1-50992 

•149 

1-05819 

•230 

1-13557 

•310 

1-23925 

•390 

1-36596 

•470 

1-51185 

•150 

1-05896 

•231 

1-13671 

•311 

1-24070 

•391 

1-36767 

•471 

1-51378 

•151 

1-05973 

•232 

1-13786 

•312 

1-24216 

•392 

1-36939 

•472 

1-51571 

•152 

1-06051 

•233 

1-13903 

•313 

1-24360 

•393 

1-37111 

•473 

1-51764 

•153 

1-06130 

•234 

1-14020 

•314 

1-24506 

•394 

1-37283 

•474 

1-51958 

•154 

1-06209 

•235 

1-14136 

•315 

1-24654 

•395 

1-37455 

•475 

1-52152 

•155 

1-06288 

•236 

1-14247 

•316 

1-24801 

•396 

1-37628 

•476 

1-52346 

•156 

1-06368- 

•237 

1-14363 

•317 

1-24946 

•397 

1-37801 

•477 

1-52541 

•157 

1-OC449 

•238 

1-14480 

•318 

1-25095 

•398 

1-37974 

•478 

1-52736 

•158 

1-06530 

•239 

1-14597 

•319 

1-25243 

•399 

1-38148 

•479 

1-52931 

•159 

1-06611 

•240 

1-14714 

•320 

1-25391 

•400 

1-38322 

•480 

1-53126 

•160 

1-06693 

•241 

1-14831 

•321 

1-25539 

•401 

1-38496 

•481 

1-53322 

•161 

1-06775 

•242 

1-14949 

•322 

1-25686 

•402 

1-38671 

•482 

1-53518 

•162 

1-06858 

•243 

1-15067 

•323 

1-25836 

•403 

1-38846 

•483 

1-53714 

•163 

1-06941 

•244 

1-15186 

•324 

1-25987 

•404 

1-39021 

•484 

1-53910 

•164 

1-07025 

•245 

1-15308 

•325 

1-26137 

•405 

1-39196 

•485 

1-54106 

•1C5 

1-07109 

•246 

1-15429 

•326 

1-26286 

•406 

1-39372 

•486 

1-54302 

•166 

1-07194 

•247 

1-15549 

•327 

1-26437 

•407 

1-39548 

•487 

1-54499 

•167 

1-07279 

•248 

1-15670 

•328 

1-26583 

•408 

1-39724 

•488 

1-54696 

•168 

1-07365 

•249 

1-15791 

•329 

1-26740 

•409 

1-39900 

•489 

1-54893 

•169 

1-07451 

•250 

1-15912 

•330 

1-26892 

•410 

1-40077 

•490 

1-55090 

•170 

1-07537 

•251 

116033 

•331 

1-27044 

•411 

1-40254 

•491 

1-55288 

•171 

1-07624 

•252 

1-16157 

•332 

1-27196 

•412 

1-40432 

•492 

1-55486 

•172 

1-07711 

•253 

1-16279 

•333 

1-27349 

•413 

1-40610 

•493 

1-55685 

•173 

1-07799 

•254 

1-16402 

•334 

1-27502 

•414 

1-40788 

•494 

1-55854 

•174 

1-07888 

•255 

1-1C526 

•335 

1-27656 

•415 

1-40966 

•495 

1-56083 

•175 

1-07977 

•256 

1-16&49 

•336 

1-27810 

•416 

1-41145 

•496 

1-56282 

•176 

1-08066 

•257 

1-16774 

•337 

1-27864 

•417 

1-41324 

•497 

1-56481 

•177 

1-08156 

•258 

1-16899 

•338 

1-28118 

•418 

1-41503 

•498 

1-56680 

•178 

1-08246 

•259 

1-17024 

•339 

1-28273 

•419 

1-41682 

•499 

1-5P879 

•179 

1-08337 

•260 

1-17150 

•340 

1-28428 

•420 

1-41861 

•500 

1-57079 

•180 

1-08428 

PROPORTIONS  OF  THE  LENGTHS  OF  SEMI-ELLIPTIC  ARCS. 


69 


PROPORTIONS    OF   THE   LENGTHS   OF   SEMI- 
ELLIPTIC   ARCS. 


Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

•100 

1-04162 

•157 

1-10113 

•214 

1-66678 

•271 

1-23835 

•328 

1.31472 

•101 

1-04262 

•158 

1-10224 

•215 

1-16799 

•272 

1-23966 

•329 

1-31610 

•102 

1-04362 

•159 

1-10335 

•216 

1-16920 

•273 

1-24097 

•330 

1-31748 

•103 

1-04462 

•160 

1-10447 

•217 

1-17041 

•274 

1-24228 

•331 

1-31886 

•104 

1-04562 

•161 

1-10560 

•218 

1-17163 

•275 

1-24359 

•332 

1-32024 

•105 

1.04662 

•162 

1-10672 

•219 

1-17285 

•276 

1-24480 

•333 

1-32162 

•106 

1-04762 

•163 

1-10784 

•220 

1-17407 

•277 

1-24612 

•334 

1-32300 

•107 

1-04862 

•164 

1-10896 

•221 

1-17529 

•278 

1-24744 

•335 

1-32438 

•108 

1-04962 

•165 

1-11008 

•222 

1-17651 

•279 

1-24876 

•336 

1-32576 

•109 

1-05063 

•166 

1-11120 

•223 

1-17774 

•280 

1-25010 

•337 

1-32715 

•110 

1-05164 

•167 

1-11232 

•224 

1-17897 

•281 

1-25142 

•338 

1-32854 

•111 

1-05265 

•168 

1-11344 

•225 

1-18020 

•282 

1-25274 

•339 

1-32993 

•112 

1-05366 

•169 

1-11456 

•226 

1-18143 

•283 

1-25406 

•340 

1-33132  • 

•113 

1-05467 

•170 

1-11569 

•227 

1-18266 

•284 

1-25538 

•341 

1-33272 

•114 

•05568 

•171 

1-11682 

•228 

1-18390 

•285 

1-25670 

•342 

1-33412 

•115 

•05669 

•172 

1-11795 

•229 

1-18514 

•286 

1-25803 

•343 

1-33552 

•116 

•05770 

•173 

1-11908 

•230 

1-18638 

•287 

1-25936 

•344 

1-33692 

•117 

•05872 

•174 

1-12021 

•231 

1-18762 

•288 

1-26069 

•345 

1-33833 

•118 

•05974 

•175 

1-12134 

•232 

1-18886 

•289 

1-26202 

•346 

1-33974 

•119 

•06076 

•176 

1-12247 

•233 

1-19010 

•290 

1-26335 

•347 

1-34115 

•120 

•06178 

•177 

1-12360 

•234 

1-19134 

•291 

1-26468 

•348 

1-34256 

•121 

1-06280 

•178 

1-12473 

•235 

1-19258 

•292 

1-26601 

•349 

1-34397 

•122 

1-06382 

•179 

1-12586 

•236 

1-19382 

•293 

1-26734 

•350 

1-34539 

•123 

1-06484 

•180 

1-12699 

•237 

1-19506 

•294 

1-26867 

•851 

1-34681 

•124 

1-06586 

•181 

1-12813 

•238 

1-19630 

•295 

1-27000 

•352 

1-34823 

•125 

•06689 

•182 

1-12927 

•239 

1-19755 

•296 

1-27133 

•353 

1-34965 

•126 

•06792 

•183 

1-13041 

•240 

1-19880 

•297 

1-27267 

•354 

1-35108 

•127 

•06895 

•184 

1-13155 

•241 

1-20005 

•298 

1-27401 

•355 

1-35251 

•128 

•06998 

•185 

1-13269 

•242 

1-20130 

•299 

1-27535 

•356 

1-35394 

•129 

•07001 

•186 

1-13383 

•243 

1-20255 

•300 

1-27669 

•357 

1-35537 

•130 

•07204 

•187 

1-13497 

•244 

1-20380 

•301 

1-27803 

•358 

1-35680 

•131 

•07308 

•188 

1-13611 

•245 

1-20506 

•302 

1-27937 

•359 

1-35823 

•132 

•07412 

•189 

1-13726 

•246 

1-20632 

•303 

1-28071 

•360 

1-35967 

•133 

•07516 

•190 

1-13841 

•247 

1-20758 

•304 

1-28205 

•361 

1-36111 

•134 

1-07621 

•191 

1-13956 

•248 

1-20884 

•305 

1-28339 

•362 

1-36255 

•135 

1-07726 

•192 

1-14071 

•249 

1-21010 

•306 

1-28474 

•363 

1-36399 

•136 

1-07831 

•193 

1-14186 

•250 

1-21136 

•307 

1-28609 

•364 

1-36543 

•137 

1-07937 

•194 

1-14301 

•251 

1-21263 

•308 

1-28744 

•365 

1-36688 

•138 

1-08043 

•195 

1-14416 

•252 

1-21390 

•309 

1-28879 

•366 

1-36833 

•139 

1-08149 

•196 

1-14531 

•253 

1-21517 

•310 

1-29014 

•367 

1-36978 

•140 

1-08255 

•197 

1-14646 

•254 

1.21644 

•311 

1-29149 

•368 

1-37123 

•141 

1-08362 

•198 

1-14762 

•255 

1-21772 

•312 

1-29285 

•369 

1-37268 

•142 

1-08469 

•199 

1-14888 

•256 

1-21900 

•313 

1-29421 

•370 

1-37414 

•143 

1-08576 

•200 

1-15014 

•257 

1-22028 

•314 

2-29557 

•371 

1-37662 

•144 

1-08684 

•201 

1-15131 

•258 

1-22156 

•315 

1-29603 

•372 

1-87708 

•145 

1-08792 

•202 

1-15248 

•259 

1-22284 

•316 

1-29829 

•373 

1-37854 

•146 

1-08901 

•203 

1-15366 

•260 

1-22412 

•317 

1-29965 

•374 

1-38000 

•147 

1-09010 

•204 

1-15484 

•261 

1-22541 

•318 

1-30102 

•375 

1-38146 

•148 

1-09119 

•205 

1-15602 

•262 

1-22670 

•319 

1-30239 

•376 

1-38292 

•149 

1-09228 

•206 

1-15720 

•263 

1-22799 

•320 

1-30376 

•377 

1-38439 

•150 

1-09330 

•207 

1-15838 

•264 

1-22928 

•321 

1-30513 

•378 

1-38585 

•151 

1-09448 

•208 

1-15957 

•265 

1-23057 

•322 

1-30650 

•379 

1-38732 

•152 

1-09558 

•209 

1-16076 

•266 

1-23186 

•323 

1-30787 

•380 

1-38879 

•153 

1-09669 

•210 

1-16196 

•267 

1-23315 

•324 

1-30924 

•381 

1-39024 

•154 

1-09780 

•211 

1-16316 

•268 

1-23445 

•325 

1-31061 

•382 

1-39169 

•155 

1-09891 

•212 

1-16436 

•269 

1-23575 

•326 

1--31198 

•383 

1-39314 

•156 

1-10002 

•213 

1-16557 

•270 

1-23705 

•327 

1-31335 

•384 

1-39459 

70 


THE   PRACTICAL   MODEL   CALCULATOR. 


Height 
of  Aro. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

•385 

•39605 

•447 

1-48850 

•509 

1-58474 

•571 

1-68195 

•633 

1-78172 

•386 

•39751 

•448 

1-49003 

•510 

1-58629 

•572 

1-68354 

•634 

1-78335 

•387 

•39897 

•449 

1-49157 

•511 

1-58784 

•573 

1-68513 

•635 

1-78498 

•388 

•40043 

•450 

1-49311 

•512 

1-58940 

•574 

1-68672 

•636 

1-78660 

•389 

•40189 

•451 

1-49465 

•513 

1-59096 

•575 

1-68831 

•637 

1-78823 

•390 

1-40335 

•452 

1-49618 

•514 

1-59252 

•576 

1-68990 

•638 

1-78986 

•391 

1-40481 

•453 

1-49771 

•515 

1-59408 

•577 

1-69149 

-639 

1-79149 

•392 

1-40627 

•454 

1-49924 

•516 

1-59564 

•578 

1-69308 

•640 

1-79312 

•393 

1-40773 

•455 

1-50077 

•517 

1-59720 

•579 

1-69467 

•641 

1-79475 

•394 

1-40919 

•456 

1-50230 

•518 

1-59876 

•580 

1-69626 

•642 

1-79638 

•395 

1-41065 

•457 

1-50383 

•519 

1-60032 

•581 

1-69785 

•643 

1-79801 

•396- 

1-41211 

•458 

1-50536 

•520 

1-60188 

•582 

1-69945 

•644 

1-79964 

•397 

1-41357 

•459 

1-50689 

•521 

1-60344 

•583 

1-70105 

•645 

1-80127 

•398 

1-41504 

•460 

1-50842 

•522 

1-60500 

•584 

1-70264 

•646 

1-80290 

•399 

1-41651 

-461 

1-50996 

•523 

1-60656 

-585 

1-70424 

•647 

1-80454 

•400 

1-41798 

•462 

1-51150 

•524 

1-60812 

•586 

1-70584 

•648 

1-80617 

•401 

1-41945 

•463 

1-51304 

•525 

1-60968 

•587 

1-70745 

•649 

1-80780 

•402 

1-42092 

•464 

1-51458 

•526 

1-61124 

•588 

1-70905 

•650 

1-80943 

•403 

1-42239 

-465 

1-51612 

•527 

1-61280 

•589 

1-71065 

•651 

1-81107 

•404 

1-42386 

•466 

1-51766 

•528 

1-61436 

•590 

1-71225 

•652 

1-81271 

•405 

1-42533 

•467 

1-51920 

•529 

1-61592 

•591 

1-71286 

•653 

1-81435 

•406 

1-42681 

•468 

1-52074 

-530 

1-61748 

•592 

1-71546 

•654 

1-81599 

•407 

1-42829 

•469 

1-52229 

•531 

1-61904 

•593 

1-71707 

•655 

1-81763 

•408 

1-42977 

•470 

1-52384 

•532 

1-62060 

-594 

1-71868 

•656 

1-81928 

•409 

1-43125 

•471 

1-52539 

•533 

1-62216 

•595 

1-72029 

•657 

1-82091 

•410 

1-43273 

.472 

1-52691 

•534 

1-62372 

•596 

1-72190 

•658 

1-82255 

•411 

1-43421 

•473 

1-52849 

•535 

1-62528 

•597 

1-72350 

•659 

1-82419 

•412 

1-43569 

.474 

1-53004 

•536 

1-62684 

•598 

1-72511 

•660 

1-82583 

•413 

1-43718 

.475 

1-53159 

.537 

1-62840 

-599 

1-72672 

•661 

1-82747 

•414 

1-43867 

•476 

1-53314 

•538 

1-62996 

•600 

1-72833 

•662 

1-82911 

•415 

1-44016 

.477 

1-53469 

•539 

1-63152 

•601 

1-72994 

-663 

1-83075 

•416 

1-44165 

.478 

1-53625 

•540 

1-63309 

•602 

1-73155 

•664 

1-83240 

•417 

1-44314 

.479 

1-53781 

•541 

1-63465 

•603 

1-73316 

•665 

1-83404 

•418 

1-44463 

.480 

1-53937 

•542 

1-63623 

•604 

1-73477 

•666 

1-83568 

•419 

1-44613 

.481 

1-54093 

•543 

1-63780 

•605 

1-73638 

•667 

1-83733 

•420 

1-44763 

.482 

1-54249 

•544 

1-63937 

•606 

1-73799 

•668 

1-83897 

•421 

1-44913 

.483 

1-54405 

•545 

1-64094 

•607 

1-73960 

•669 

1-84061 

•422 

1-45064 

.484 

1-54561 

•546 

1-64251 

•608 

1-74121 

•670 

1-84226 

•423 

1-45214 

.485 

1-54718 

•547 

1-64408 

•609 

1-74283 

•671 

1-84391 

•424 

1-45364 

.486 

1-54875 

•548 

1-64565 

•610 

1-74444 

-672 

1-84556 

•425 

1-45515 

.487 

1-55032 

•549 

1-64722 

-611 

1-74605 

•673 

1-84720 

•426 

1-45665 

.488 

1-55189 

•550 

1-64879 

•612 

1-74767 

•674 

1-84885 

•427 

1-45815 

.489 

1-55346 

•551 

1-65036 

-613 

1-74929 

•675 

1  -85050 

•428 

1-45966 

•490 

1-55503 

•552 

1-65193 

•614 

1-75091 

•676 

1-85215 

•429 

1-46167 

•491 

1-55660 

-553 

1-65350 

-615 

1-75252 

•677 

1  -85379 

•430 

1-46268 

•492 

1-55817 

•554 

1-65507 

-616 

1-75414 

•678 

1-85544 

•431 

1-46419 

•493 

1-55974 

-555 

1-65665 

-617 

1-75576 

•679 

1-85709 

•432 

1-46570 

494 

1-56131 

•556 

1-65823 

•618 

1-75738 

•680 

1-85874 

•433 

1-46721 

495 

1-56289 

•557 

1-65981 

•619 

1-75900 

•681 

1-86039 

•434 

1-46872 

496 

1-56447 

•558 

1-66139 

•620 

1-76062 

•682 

1-86205 

•435 

1-47023 

•497 

1-56605 

•559 

1-66297 

•621 

1-76224 

•683 

1-86370 

•436 

1-47174 

498 

1-56763 

•560 

1-66455 

•622 

1-76386 

•684 

1  -86535 

•437 

1-47326 

499 

1-56921 

•561 

1-66613 

•623 

1-76548 

•685 

1-86700 

•438 

1-47478 

•500 

1-57089 

•562 

1-66771 

•624 

1-76710 

•686 

1-86866 

•439 

1-47630 

•501 

1-57234 

•563 

1-66929 

•625 

1-76872 

•687 

1-87031 

•440 

1-47782 

•502 

1-57389 

•564 

1-67087 

•626 

1-77034 

-688 

1-87196 

•441 

1-47934 

•503 

1-57544 

•565 

1-67245 

•627 

1-77197 

•689 

1  -87362 

•442 

1-48086 

•504 

1-57699 

•566 

1-67403 

•628 

1-77359 

•690 

1-87527 

•443 

1-48238 

•505 

1-57854 

•567 

1-67561 

•629 

1-77521 

•691 

1-87693 

•444 

1-48391 

•506 

1-58009 

•568 

1-67719 

•630 

1-77684 

•692 

1-87859 

•445 

1-48544 

-507 

1-58164 

•569 

1-67877 

•631 

1-77847 

•693 

1-88024 

•446 

1-48697 

508 

1-58319 

•570 

1-68036 

•632 

1-78009 

-694 

1-88190 

PROPORTIONS  OF  THE  LENGTHS  OF  SEMI-ELLIPTIC  ARCS.          71 


Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  of 
Arc. 

Height 
of  Arc. 

Length  oi 
Arc. 

Height  j  Length  of 
of  Arc.          Arc. 

•695 

1-88356 

•757 

1-98794 

•818 

2-09360 

•879 

2-20292 

•940 

2-31479 

•696 

1-88522 

•758 

1-98964 

•819 

2-09536 

•880 

2-20474 

•941 

2-31666 

•697 

1-88688 

•759 

1-99134 

•820 

2-09712 

•881 

2-20656 

•942 

2-31852 

•698 

1-88854 

•760 

1-99305 

•821 

2-09888 

•882 

2-20839 

•943 

2-32038 

•699 

1-89020 

•761 

1-99476 

•822 

2-10065 

•883 

2-21022 

•944 

2-32224 

•700 

1-89186 

•762 

1-99647 

•823 

2-10242 

•884 

2-21205 

•945 

2-32411 

•701 

1-89352 

•763 

1-99818 

•824 

2-10419 

•885 

2-21388 

•946 

2-32598 

•702 

1-89519 

•764 

1-99989 

•825 

2-10596 

•886 

2-21571 

•947 

2-32785 

•703 

1-89685 

•765 

2-00160 

•826 

2-10773 

•887 

2-21754 

•948 

2-32972 

•704 

1-89851 

•766 

2-00331 

•827 

2-10950 

•888 

2-21937 

•949 

2-33160 

•705 

1-90017 

•767 

2-00502 

•828 

2-11127 

•889 

2-22120 

•950 

2-33348 

•706 

1-90184 

•768 

2-00673 

•829 

2-11304 

•890 

2-22303 

•951 

2-33537 

•707 

1-90350 

•769 

2-00844 

•830 

2-11481 

•891 

2-22486 

•952 

2-33726 

•708 

1-90517 

•770 

2-01016 

•831 

2-11659 

•892 

2-22670 

•953 

2-33915 

•709 

1-90684 

•771 

2-01187 

•832 

2-11837 

•893 

2-22854 

•954 

2-34104 

•710 

1-90852 

•772 

2-01350 

•833 

2-12015 

•894 

2-23038 

•955 

2-34293 

•711 

1-91019 

•773 

2-01531 

•834 

2-12193 

•895 

2-23222 

•956 

2-34483 

•712 

1-91187 

•774 

2-01702 

•835 

2-12371 

•896 

2-23406 

•957 

2-34673 

•713 

1-91355 

•775 

2-01874 

•836 

2-12549 

•897 

2-23590 

•958 

2-34862 

•714 

1-91523 

•776 

2-02045 

•837 

2-12727 

•898 

2-23774 

•959 

2-35051 

•715 

1-91691 

•777 

2-02217 

•838 

2-12905 

•899 

2-23958 

•960 

2-35241 

•716 

1-91859 

•778 

2-02389 

•839 

2-13083 

•900 

2-24142 

•961 

2-35431 

•717 

1-92027 

•779 

2-02561 

•840 

2-13261 

•901 

2-24325 

•962 

2-35621 

•718 

1-92195 

•780 

2-02733 

•841 

2-13439 

•902 

2-24508 

•963 

2-35810 

•719 

1-92363 

•781 

2-02907 

•842 

2-13618 

•903 

2-24691 

•964 

2-36000 

•720 

1-92531 

•782 

2-03080 

•843 

2-13797 

•904 

2.24874 

•965 

2-36191 

•721 

1-92700 

•783 

2-03252 

•844 

2-13976 

•905 

2-25057 

•966 

2-36381 

•722 

1-92868 

•784 

2-03425 

•845 

2-14155 

•906 

2-25240 

•967 

2-36571 

•723 

1-93036 

•785 

2-03598 

•846 

2-14334 

•907 

2-25423 

•968 

2-36762 

•724 

1-93204 

•786 

2-03771 

•847 

2-14513 

•908 

2-25606 

•969 

2-36952 

•725 

1-93373 

•787 

2-03944 

•848 

2-14692 

•909 

2-25789 

•970 

2-37143 

•726 

1-93541 

•788 

2-04117 

•849 

2-14871 

•910 

2-25972 

•971 

2-37334 

•727 

1-93710 

•789 

2-04290 

•850 

2-15050 

•911 

2-26155 

•972 

2-37525 

•728 

1-93878 

•790 

2-04462 

•851 

2-15229 

•912 

2-26338 

•973 

2-37716 

•729 

1-94046 

•791 

2-04635 

•852 

2-15409 

•913 

2-26521 

•974 

2-37908 

•730 

1-94215 

•792 

2-04809 

•853 

2-15589 

•914 

2-26704 

•975 

2-38100 

•731 

1-94383 

•793 

2-04983 

•854 

2-15770 

•915 

2-26888 

•976 

2-38291 

•732 

1-94552 

•794 

2-05157 

•855 

2-15950 

•916 

2-27071 

•977 

2-38482 

•733 

1-94721 

•795 

2-05331 

•856 

2-16130 

•917 

2-27254 

•978 

2-38673 

•734 

1-94890 

•796 

2-05505 

•857 

2-16309 

•918 

2-27437 

•979 

2-38864 

•735 

1-95059 

•797 

2-05679 

•858 

2-16489 

•919 

2-27620 

•980 

2-39055 

•736 

1-95228 

•798 

2-05853 

•859 

2-16668 

•920 

2-27803 

•981 

2-39247 

•737 

1-95397 

•799 

2-06027 

•860 

2-16848 

•921 

2-27987 

•982 

2-39439 

•738 

1-95566 

•800 

2-06202 

•861 

2-17028 

•922 

2-28170 

•983 

2-39631 

•739 

1-95735 

•801 

2-06377 

•862 

2-17209 

•923 

2-28354 

•984 

2-39823 

•740 

1-95994 

•802 

2-06552 

•863 

2-17389 

•924 

2-28537 

•985 

2-40016 

•741 

1-96074 

•803 

2-06727 

•864 

2-17570 

•925 

2-28720 

•986 

2-40208 

•742 

1-96244 

•804 

2-06901 

•865 

2-17751 

•926 

2-28903 

•987 

2-40400 

•743 

1-96414 

•805 

2-07076 

•866 

2-17932 

•927 

2-29086 

•988 

2-40592 

•744 

1-96583 

•806 

2-07251 

•867 

2-18113 

•928 

2-29270 

•989 

2-40784 

•745 

1-96753 

•807 

2-07427 

•868 

2-18294 

•929 

2-29453 

•990 

2-40976 

•746 

1-96923 

•808 

2-07602 

•869 

2-18475 

•930 

2-29636 

•991 

2-41169 

•747 

1-97093 

•809 

2-07777 

•870 

2-18656 

•931 

2-29820 

•992 

2-41362 

•748 

1-97262 

•810 

2-07953 

•871 

2-18837 

•932 

2-30004 

•993 

2-41556 

•749 

1-97432 

•811 

2-08128 

•872 

2-19018 

•933 

2-30188 

•994 

2-41749 

•750 

1-97602 

•812 

2-08304 

•873 

2-19200 

•934 

2-30373 

•995 

2-41943 

•751 

1-97772 

•813 

2-08480 

•874 

2-19882 

•935 

2-30557 

•996 

2-42136 

•752 

1-97943 

•814 

2-08656 

•875 

2-19564 

•936 

2-30741 

•997 

2-42329 

•753 

1-98113 

•815 

2-08832 

•876 

2-19746 

•937 

2-30926 

•998 

2-42522 

-754 

1-98283 

•816 

2-09008 

•877 

2-19928 

•938 

2-31111 

•999 

2-42715 

•755 

1-98453 

•817 

2-09198 

•878 

2-20110 

•939 

2-31295 

1000 

2-42908 

•756 

1-98623 

72  THE  PRACTICAL  MODEL  CALCULATOR. 

To  find  the  length  of  an  arc  of  a  circle,  or  the  curve  of  a  right 
semi-ellipse. 

RULE. — Divide  the  height  by  the  base,  and  the  quotient  will  be 
the  height  of  an  arc  of  which  the  base  is  unity.  Seek,  in  the 
Table  of  Circular  or  of  Semi-elliptical  arcs,  as  the  case  may  be, 
for  a  number  corresponding  to  this  quotient,  and  take  the  length 
of  the  arc  from  the  next  right-hand  column.  Multiply  the  number 
thus  taken  out  by  the  base  of  the  arc,  and  the  product  will  be  the 
length  of  the  arc  or  curve  required. 

In  a  Bridge,  suppose  the  profiles  of  the  arches  are  the  arcs  of 
circles ;  the  span  of  the  middle  arch  is  240  feet  and  the  height  24  feet ; 
required  the  length  of  the  arc. 

24  -f-  240  =  -100 ;  and  -100,  by  the  Table,  is  1-02645. 
Hence  1-02645  X  24  =  246-34800  feet,  the  length  required. 
The  profiles  of  the  arches  of  a  Bridge  are  all  equal  and  similar 
semi-ellipses ;  the  span  of  each  is  120  feet,  and  the  rise  18  feet ; 
required  the  length  of  the  curve. 

28  -f-  120  =  -233  ;  and  -233  by  the  Table,  is  1-19010. 
Hence  1-19010  X  120  =  142-81200  feet,  the  length  required. 

In  this  example  there  is,  in  the  division  of  28  by  120,  a  remainder 
of  40,  or  one-third  part  of  the  divisor ;  consequently,  the  answer, 
142-81200,  is  rather  less  than  the  truth.  But  this  difference,  in 
even  so  large  an  arch,  is  little  more  than  half  an  inch ;  therefore, 
except  where  extreme  accuracy  is  required,  it  is  not  worth  com- 
puting. 

These  Tables  are  equally  useful  in  estimating  works  which  may 
be  carried  into  practice,  and  the  quantity  of  work  to  be  executed 
from  drawings  to  a  scale. 

As  the  Tables  do  not  afford  the  means  of  finding  the  lengths  of 
the  curves  of  elliptical  arcs  which  are  less  than  half  of  the  entire 
figure,  the  following  geometrical  method  is  given  to  supply  the 
defect.^ 

Let  the  curve,  of  which  the  length  is  required  to  be  found,  be 
ABC. 

fc 


g 

Produce  the  height  line  Bd  to  meet  the  centre  of  the  curve 
in  g.  Draw  the  right  line  A^,  and  from  the  centre  #,  with  the 
distance  #B  describe  an  arc  BA,  meeting  Ag  in  h.  Bisect  Ah 
in  i,  and  from  the  centre  g  with  the  radius  gi  describe  the  arc  ik, 
meeting  dR  produced  to  k ;  then  ik  is  half  the  arc  ABC. 


TABLE    OF   RECIPROCALS    OP   NUMBERS. 


A  TABLE  of  the  Reciprocals  of  Numbers;  or  the  DECIMAL  FRAC- 
TIONS corresponding  to  VULGAR  FRACTIONS  of  which  the  Numera- 
tor is  unity  or  1. 

[In  the  following  Tables,  the  Decimal  fractions  are  Reciprocals 
of  the  Denominators  of  those  opposite  to  them ;  and  their  product 
unity. 


is 


To  find  the  Decimal  corresponding  to  a  fraction  having  a  higher 
Numerator  than  1,  multiply  the  Decimal  opposite  to  the  given  De- 
nominator, by  the  given  Numerator.  Thus,  the  Decimal  corre- 
sponding to  ei  being  -015625,  the  Decimal  to  J|  will  be  -015625  X 
15  =  -234375.] 


Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

1/2 

•5 

1/47 

•0212766 

1/92 

•010869565 

1/3 

•333333333 

1/48 

•020833333 

1/93 

•010752688 

1/4 

•25 

1/49 

•020408163 

1/94 

•010638298 

1/5 

•2 

1/50 

•02 

1/95 

•010526316 

1/6 

•166666667 

1/51 

•019607843 

1/96 

•010416667 

1/7 

•142857143 

1/52 

•019230769 

1/97 

•010309278 

1/8 

•125 

1/53 

•018867925 

1/98 

•010204082 

1/9 

•111111111 

1/54 

•018518519 

1/99 

•01010101 

1/10 

•1 

1/55 

•018181818 

1/100 

•01 

1/11 

•090909091 

1/56 

•017857143 

1/101 

•00990099 

1/12 

•083333333 

1/57 

•01754386 

1/102 

•009803922 

1/13 

•076923077 

1/58 

•017241379 

1/103 

•009708738 

1/14 

•071428571 

1/59 

•016949153 

1/104 

•009615385* 

1/15 

•066666667 

1/60 

•016666667 

1/105 

•00952381 

1/16 

•0625 

1/61 

•016393443 

1/106 

•009433962 

1/17 

•058823529 

1/62 

•016129032 

1/107 

•009345794 

1/18 

•055555556 

1/63 

•015873016 

1/108 

•009259259 

1/19 

•052631579 

1/64 

•015625 

1/109 

•009174312 

1/20 

•05 

1/65 

•015384615 

1/110 

•009090909 

1/21 

•047619048 

1/66 

•015151515 

1/111 

•009009009 

1/22 

•045454545 

1/67 

•014925373 

1/112 

•008928571 

1/23 

•043478261 

1/68 

•014705882 

1/113 

•008849558 

1/24 

•041666667 

1/69 

•014492754 

1/114 

•00877193 

1/25 

•04 

1/70 

•014285714 

1/115 

•008695652 

1/26 

•038461538 

1/71 

•014084517 

1/116 

•00892069 

1/27 

•037037037 

1/72 

•013888889 

1/117 

•008547009 

1/28 

•035714286 

1/73 

•01369863 

1/118 

•008474576 

1/29 

•034482759 

1/74 

•013513514 

1/119 

•008403361 

1/30 

•033333333 

1/75 

•013333333 

1/120 

•008333333 

1/31 

•032258065 

1/76 

•013157895 

1/121 

•008264463 

1/32 

•03125 

1/77 

•012987013 

1/122 

•008196721 

1/33 

•030303030 

1/78 

•012820513 

1/123 

•008130081 

1/34 

•029411765 

1/79 

•012658228 

1/124 

•008064516 

1/35 

•028571429 

1/80 

•0125 

1/125 

•008 

1/36 

•027777778 

1/81 

•012345679 

1/126 

•007936508 

1/37 

•027027027 

1/82 

•012195122 

1/127 

•007874016 

1/38 

•026315789 

1/83 

•012048193 

1/128 

•0078125 

1/39 

•025641026 

1/84 

•011904762 

1/129 

•007751938 

1/40 

•025 

1/85 

•011764706 

1/130 

•007692308 

1/41 

•024390244 

1/86 

•011627907 

1/131 

•007633588 

1/42 

•023809524 

1/87 

•011494253 

1/132 

•007575758 

1/43 

•023255814 

1/88 

•011363636 

1/133 

•007518797 

1/44 

•022727273 

1/89 

•011235955 

1/134 

•007462687 

1/45 

•022222222 

1/90 

•011111111 

1/135 

•007407407 

1/46 

•02173913 

1/91 

•010989011 

1/136 

•007352941 

THE   PRACTICAL   MODEL   CALCULATOR. 


Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 

Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

1/137 

•00729927 

1/198 

•005050505 

1/259 

•003861004 

1/138 

•007246377 

1/199 

•005025126 

1/260 

•003846154 

1/139 

•007194245 

1/200 

•005 

1/261 

•003831418 

1/140 

•007142857 

1/201 

•004975124 

1/262 

•003816794 

1/141 

•007092199 

1/202 

•004950495 

1/263 

•003802281 

1/142 

•007042254 

1/203 

•004926108 

1/264 

•003787879 

1/143 

•006993007 

1/204 

•004901961 

1/265 

•003773585 

1/144 

•006944444 

1/205 

•004878049 

1/266 

•003759398 

1/145 

•006896552 

1/206 

•004854369 

1/267 

•003745318 

1/146 

•006849315 

1/207 

•004830918 

1/268 

•003731343 

1/147 

•006802721 

1/208 

•004807692 

1/269 

•003717472 

1/148 

•006756757 

1/209 

•004784689 

1/270 

•003703704 

1/149 

•006711409 

1/210 

•004761905 

1/271 

•003690037 

1/150 

•006666667 

1/211 

•004739336 

1/272 

•003676471 

1/151 

•006622517 

1/212 

•004716981 

1/273 

•003663004 

1/152 

•006578947 

1/213 

•004694836 

1/274 

•003649635 

1/153 

•006535948 

1/214 

•004672897 

1/275 

•003636364 

1/154 

•006493506 

1/215 

•004651163 

1/276 

•003623188 

1/155 

•006451613 

1/216 

•00462963 

1/277 

•003610108 

1/156 

•006410256 

1/217 

•004608295 

1/278 

•003597122 

1/157 

•006369427 

1/218 

•004587156 

1/279 

•003584229 

1/158 

•006329114 

1/219 

•00456621 

1/280 

•003571429 

1/159 

•006289308 

1/220 

•004545455 

1/281 

•003558719 

1/160 

•00625 

1/221 

•004524887 

1/282 

•003546099 

1/161 

•00621118 

1/222 

•004504505 

1/283 

•003533569 

1/162 

•00617284 

1/223 

•004484305 

1/284 

•003522127 

1/163 

•006134969 

1/224 

•004464286 

1/285 

•003508772 

1/164 

•006097561 

1/225 

•004444444 

1/286 

•003496503 

1/165 

•006060606 

1/226 

•004424779 

1/287 

•003484321 

1/166 

•006024096 

1/227 

•004405286 

1/288 

•003472222 

1/167 

•005988024 

1/228 

•004385965 

1/289 

•003460208 

1/168 

•005952381 

1/229 

•004366812 

1/290 

•003448276 

1/169 

•00591716 

1/230 

•004347826 

1/291 

•003436426 

1/170 

•005882353 

1/231 

•004329004 

1/292 

•003424658 

1/171 

•005847953 

1/232 

•004310345 

1/293 

•003412969 

1/172 

•005813953 

1/233 

•004291845 

1/294 

•003401361 

1/173 

•005780347 

1/234 

•004273504 

1/295 

•003389831 

1/174 

•005747126 

1/235 

•004255319 

1/296 

•003378378 

1/175 

•005714286 

1/236 

•004237288 

1/297 

•003367003 

1/176 

•005681818 

1/237 

•004219409 

1/298 

•003355705 

1/177 

•005649718 

1/238 

•004201681 

1/299 

•003344482 

1/178 

•005617978 

1/239 

•0041841 

1/300 

•003333333 

1/179 

•005586592 

1/240 

•004166667 

1/301 

•003322259 

1/180 

•005555556 

1/241 

•004149378 

1/302 

•003311258 

1/181 

•005524862 

1/242 

•004132231 

1/303 

•00330133 

1/182 

•005494505 

1/243 

•004115226 

1/304 

•003289474 

1/183 

•005464481 

1/244 

•004098361 

1/305 

•003278689 

1/184 

•005434783 

1/245 

•004081633 

1/306 

•003267974 

1/185 

•005405405 

1/246 

•004065041 

1/307 

•003257329 

1/186 

•005376344 

1/247 

•004048583 

1/308 

•003246753 

1/187 

•005347594 

1/248 

•004032258 

1/309 

•003236246 

1/188 

•005319149 

1/249 

•004016064 

1/310 

•003225806 

1/189 

•005291005 

1/250 

•004 

1/311 

•003215434 

1/190 

•005263158 

1/251 

•003984064 

1/312 

•003205128 

1/191 

•005235602 

1/252 

•003968254 

1/313 

•003194888 

1/192 

•005208333 

1/253 

•003952569 

1/314 

•003184713 

1/193 

•005181347 

1/254 

•003937008 

1/315 

•003174603 

1/T94 

•005154639 

1/255 

•003921569 

1/316 

•003164557 

1/195 

•005128205 

1/256 

•00390625 

1/317 

•003154574 

1/196 

•005102041 

1/257 

•003891051 

1/318 

•003144654 

1/197 

•005076142 

1/258 

•003875969 

1/319 

•003134796 

TABLE    OF   RECIPROCALS    OP   NUMBERS. 


Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

1/320 

•003125 

1/381 

•002624672 

1/442 

•002262443 

1/321 

•003115265 

1/382 

•002617801 

1/443 

•002257336 

1/322 

•00310559 

1/383 

•002610966 

1/444 

•002252252 

1/323 

•003095975 

1/384 

•002604167 

1/445 

•002247191 

1/324 

•00308642 

1/385 

•002597403 

1/446 

•002242152 

1/325 

•003076923 

1/386 

•002590674 

1/447 

•002237136 

1/326 

•003067485 

1/387 

•002583979 

1/448 

•002232143 

1/327 

•003058104 

1/388 

•00257732 

1/449 

•002227171 

1/328 

•00304878 

1/389 

•002570694 

1/450 

•002222222 

1/329 

•003039514 

1/390 

•002564103 

1/451 

•002217295 

1/330 

•003030303 

1/391 

•002557545 

1/452 

•002212389 

1/331 

•003021148 

1/392 

•00255102 

1/453 

•002207506 

1/332 

•003012048 

1/393 

•002544529 

1/454 

•002202643 

1/333 

•003003003 

1/394 

•002538071 

1/455 

•002197802 

1/334 

•002994012 

1/395 

•002531646 

1/456 

•002192982 

1/335 

•002985075 

1/396 

•002525253 

1/457 

•002188184 

1/336 

•00297619 

1/397 

•002518892 

1/458 

•002183406 

1/337 

•002967359 

1/398 

•002512563 

1/459 

•002178649 

1/338 

•00295858 

1/399 

•002506266 

1/460 

•002173913 

1/339 

•002949853 

1/400 

•0025 

1/461 

•002169197 

1/340 

•002941176 

1/401 

•002493766 

1/462 

•002164502 

1/341 

•002932551 

1/402 

•002487562 

1/463 

•002159827 

1/342 

•002923977 

1/403 

•00248139 

1/464 

•002155172 

1/343 

•002915452 

1/404 

•002475248 

1/465 

•002150538 

1/344 

•002906977 

1/405 

•002469136 

1/466 

•002145923 

1/345 

•002898551 

1/406 

•002463054 

1/467 

•002141328 

1/346 

•002890173 

1/407 

•002457002 

1/468 

•002136752 

1/347 

•002881844 

1/408 

•00245098 

1/469 

•002132196 

1/348 

•002873563 

1/409 

•002444988 

1/470 

•00212766 

1/349 

•00286533 

1/410 

•002439024 

1/471 

•002123142 

1/350 

•002857143 

1/411 

•00243309 

1/472 

•002118644 

1/351 

•002849003 

1/412 

•002427184 

1/473 

•002114165 

1/352 

•002840909 

1/413 

•002421308 

1/474 

•002109705 

1/353 

•002832861 

1/414 

•002415459 

1/475 

•002105263 

1/354 

•002824859 

1/415 

•002409639 

1/476 

•00210084 

1/355 

•002816901 

1/416 

•002406846 

1/477 

•002096486 

1/356 

•002808989 

1/417 

•002398082 

1/478 

•00209205 

1/357 

•00280112 

1/418 

•002392344 

1/479 

•002087683 

1/358 

•002793296 

1/419 

•002386635 

1/480 

•002083333 

1/359 

•002785515 

1/420 

•002380952 

1/481 

•002079002 

1/360 

•002777778 

1/421 

•002375297 

1/482 

•002074689 

1/361 

•002770083 

1/422 

•002369668 

1/483 

•002070393 

1/362 

•002762431 

1/423 

•002364066 

1/484 

•002066116 

1/363 

•002754821 

1/424 

•002358491 

1/4.85 

•002061856 

1/364 

•002747235 

1/425 

•002352941 

1/486 

•002057613 

1/365 

•002739726 

1/426 

•002347418 

1/487 

•002053388 

1/366 

•00273224 

1/427 

•00234192 

1/488 

•00204918 

1/367 

•002724796 

1/428 

•002336449 

1/489 

•00204499 

1/368 

•002717391 

1/429 

•002331002 

1/490 

•002040816 

1/369 

•002710027 

1/430 

•002325581 

1/491 

•00203666 

1/370 

•002702703 

1/431 

•002320186 

1/492 

•00203252 

1/371 

•002695418 

1/432 

•002314815 

1/493 

•002028398 

1/372 

•002688172 

1/433 

•002309469 

1/494 

•002024291 

1/373 

•002680965 

1/434 

•002304147 

1/495 

•002020202 

1/374 

•002673797 

1/435 

•002298851 

1/496 

•002016129 

1/375 

•002666667 

1/436 

•002293578 

1/497 

•002012072 

1/376 

•002659574 

1/437 

•00228833 

1/498 

•002008032 

1/377 

•00265252 

1/438 

•002283105 

1/499 

•002004008 

1/378 

•002645503 

1/439 

•002277904 

1/500 

•002 

1/379 

•002638521 

1/440 

•002272727 

1/501 

•001996008 

1/380 

•002631579 

1/441 

•002267574 

1/502 

•001992032 

76 


THE    PRACTICAL    MODEL    CALCULATOR. 


Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Nuinb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

1/503 

•001988072 

1/564 

•00177305 

1/625 

•0016 

1/504 

•001984127 

1/565 

•001769912 

1/626 

•001597444 

1/505 

•001980198 

1/566 

•001766784 

1/627 

•001594896 

1/506 

•001976285 

1/567 

•001763668 

1/628 

•001592357 

1/507 

•001972387 

1/568 

•001760563 

1/629 

•001589825 

1/508 

•001968504 

1/569 

•001757469 

1/630 

•001587302 

1/509 

•001964637 

1/570 

•001754386 

1/631 

•001584786 

1/510 

•001960784 

1/571 

•001751313 

1/632 

•001582278 

1/511 

•001956947 

1/572 

•001748252 

1/633 

•001579779 

1/512 

•001953125 

1/573 

•001745201 

1/634 

•001577287 

1/513 

•001949318 

1/574 

•00174216 

1/635 

•001574803 

1/514 

•001945525 

1/575 

•00173913 

1/636 

•001572327 

1/515 

•001941748 

1/576 

•001736111 

1/637 

•001569859 

1/516 

•001937984 

1/577 

•001733102 

1/638 

•001567398 

1/517 

•001934236 

1/578 

•001730104 

1/639 

•001564945 

1/518 

•001930502 

1/579 

•001727116 

1/640 

•0015625 

1/519 

•001926782 

1/580 

•001724138 

1/641 

•001560062 

1/520 

•001923077 

1/581 

•00172117 

1/642 

•001557632 

1/521 

•001919386 

1/582 

•001718213 

1/643 

•00155521 

1/522 

•001915709 

1/583 

•001715266 

1/644 

•001552795 

1/523 

•001912046 

1/584 

•001712329 

1/645 

•001550388 

1/524 

•001908397 

1/585 

•001709402 

1/646 

•001547988 

1/525 

•001904762 

1/586 

•001706485 

1/647 

•001545595 

1/526 

•001901141 

1/587 

•001703578 

1/648 

•00154321 

1/527 

•001897533 

1/588 

•00170068 

1/649 

•001540832 

1/528 

•001893939 

1/589 

•001697793 

1/650 

•001538462 

1/529 

•001890359 

1/590 

•001694915 

1/651 

•001536098 

1/530 

•001886792 

1/591 

•001692047 

1/652 

•001533742 

1/531 

•001883239 

1/592 

•001689189 

1/653 

•001531394 

1/532 

•001879699 

1/593 

•001686341 

1/654 

•001529052 

1/533 

•001876173 

1/594 

•001683502 

1/655 

•001526718 

1/534 

•001872659 

1/595 

•001680672 

1/656 

•00152439 

1/535 

•001869159 

1/596 

•001677852 

1/657 

•00152207 

1/536 

•001865672 

1/597 

•001675042 

1/658 

•001519751 

1/537 

•001862197 

1/598 

•001672241 

1/659 

•001517451 

1/538 

•001858736 

1/599 

•001669449 

1/660 

•001515152 

1/539 

•001855288 

1/600 

•001666667 

1/661 

•001512859 

1/540 

•001851852 

1/601 

•001663894 

1/662 

•001510574 

1/541 

•001848429 

1/602 

•00166113 

1/663 

•001508296 

1/542 

•001845018 

1/603 

•001658375 

1/664 

•001506024 

1/543 

•001841621 

1/604 

•001655629 

1/665 

•001503759 

1/544 

•001838235 

1/605 

•001652893 

1/666 

•001501502 

1/545 

•001834862 

1/606 

•001650165 

1/667 

•00149925 

1/546 

•001831502 

1/607 

•001647446 

1/668 

•001497006 

1/547 

•001828154 

1/608 

•001644737 

1/669 

•001494768 

1/548 

•001824818 

1/609 

•001642036 

1/670 

•001492537 

1/549 

•001821494 

1/610 

•001639344 

1/671 

•001490313 

1/550 

•001818182 

1/611 

•001636661 

1/672 

•001488095 

1/551 

•001814882 

1/612 

•001633987 

1/673 

•001485884 

1/552 

•001811594 

1/613 

•001631321 

1/674 

•00148368 

1/553 

•001808318 

1/614 

•001628664 

1/675 

•001481481 

1/554 

•001805054 

1/615 

•001626016 

1/676 

•00147929 

1/555 

•001801802 

1/616 

•001623377 

1/677 

•001477105 

1/556 

•001798561 

1/617 

•001620746 

1/678 

•001474926 

1/557 

•001795332 

1/618 

•001618123 

1/679 

•001472754 

1/558 

•001792115 

1/619 

•001615509 

1/680 

•001470588 

1/559 

•001788909 

1/620 

•001612903 

1/681 

•001468429 

1/560 

•001785714 

1/621 

•001610306 

1/682 

•001466276 

1/561 

•001782531 

1/622 

•001607717 

1/683 

•001464129 

1/562 

•001779359 

1/623 

•001605136 

1/684 

•001461988 

1/563 

•001776199 

1/624 

•001602564 

1/685 

•001459854 

TABLE    OF   RECIPROCALS    OF   NUMBERS. 


77 


Fraction  or 

Decimal  or 

Fraction  or 

Decimal  or 

Fraction  or 

Decimal  or 

Nurnb. 

Reciprocal. 

Numb. 

Reciprocal. 

Numb. 

Reciprocal. 

1/686 

•001457726 

1/747 

•001338688 

1/808 

•001237624 

1/687 

•001455604 

1/748 

•001336898 

1/809 

•001236094 

1/688 

•001453488 

1/749 

•001335113 

1/810 

•001234568 

1/689 

•001451379 

1/750 

•001333333 

1/811 

•001233046 

1/690 

•001449275 

1/751 

•001331558 

1/812 

•001231527 

1/691 

•001447178 

1/752 

•001329787 

1/813 

•001230012 

1/692 

•001445087 

1/753 

•001328021 

1/814 

•001228501 

1/693 

•001443001 

1/754 

•00132626 

1/815 

•001226994 

1/694 

•001440922 

1/755 

•001324503 

1/816 

•001225499 

1/695 

•001438849 

1/756 

•001322751 

1/817 

•00122399 

1/696 

•001436782 

1/757 

•001321004 

1/818 

•001222494 

1/697 

•00143472 

1/758 

•001319261 

1/819 

•001221001 

1/698 

•001432665 

1/759 

•001317523 

1/820 

•001219512 

]/699 

•001430615 

1/760 

•001315789 

1/821 

•001218027 

1/700 

•001428571 

1/761 

•00131406 

1/822 

•001216545 

1/701 

•001426534 

1/762 

•001312336 

1/823 

•001215067 

1/702 

•001424501 

1/763 

•001310616 

1/824 

•001213592 

1/703 

•001422475 

1/764 

•001308901 

1/825 

•001212121 

1/704 

•001420455 

1/765 

•00130719 

1/826 

•001210654 

1/705 

•00141844 

1/766 

•001305483 

1/827 

•00120919 

1/706 

•001416431 

1/767 

•001303781 

1/828 

•001207729 

1/707 

•001414427 

1/768 

•001302083 

1/829 

•001206273 

1/708 

•001412429 

1/769 

•00130039 

1/830 

•001204819 

1/709 

•001410437 

1/770 

•001298701 

1/831 

•001203369 

1/710 

•001408451 

1/771 

•001297017 

1/832 

•001201923 

1/711 

•00140647 

1/772 

•001295337 

1/833 

•00120048 

1/712 

•001404494 

1/773 

•001293661 

1/834 

•001199041 

1/713 

•001402525 

1/774 

•00129199 

1/835 

•001197605 

1/714 

•00140056 

1/775 

•001290323 

1/836 

•001196172 

1/715 

•001398601 

1/776 

•00128866 

1/837 

•001194743 

1/716 

•001396648 

1/777 

•001287001 

1/838 

•001193317 

1/717 

•0013947 

1/778 

•001285347 

1/839 

•001191895 

1/718 

•001392758 

1/779 

•001283697 

1/840 

•001190476 

1/719 

•001390821 

1/780 

•001282051 

1/841 

•001189061 

1/720 

•001388889 

1/781 

•00128041 

1/842 

•001187648 

1/721 

•001386963 

1/782 

•001278772 

1/843 

•00118624 

1/722 

•001385042 

1/783 

•001277139 

1/844 

•001184834 

1/723 

•001383126 

1/784 

•00127551 

1/845 

•001183432 

1/724 

•001381215 

1/785 

•001273885 

1/846 

•001182033 

1/725 

•00137931 

1/786 

•001272265 

1/847 

•001180638 

1/726 

•00137741 

1/787 

•001270648 

1/848 

•001179245 

1/727 

•001375516 

1/788 

•001269036 

1/849 

•001177856 

1/728 

•001373626 

1/789 

•001267427 

1/850 

•001176471 

1/729 

•001371742 

1/790 

•001265823 

1/851 

•001175088 

1/730 

•001369863 

1/791 

•001264223 

1/952 

•001173709 

1/731 

•001367989 

1/792 

•001262626 

1/853 

•001172333 

1/732 

•00136612 

1/793 

•001261034 

1/854 

•00117096 

1/733 

•001364256 

1/794 

•001259446 

1/855 

•001169591 

1/734 

•001362398 

1/795 

•001257862 

1/856 

•001168224 

1/735 

•001360544 

1/796 

•001256281 

1/857 

•001166861 

1/736 

•001358696 

1/797 

•001254705 

1/858 

•001165501 

1/737 

•001356852 

1/798 

•001253133 

1/859 

•001164144 

1/738 

•001355014 

1/799 

•001251364 

1/860 

•001162791 

1/739 

•00135318 

1/800 

•00125 

1/861 

•00116144 

1/740 

•001351351 

1/801 

•001248439 

1/862 

•001160093 

1/741 

•001349528 

1/802 

•001246883 

1/863 

•001158749 

1/742 

•001347709 

1/803 

•00124533 

1/864 

•001157407 

1/743 

•001345895 

1/804 

•001243781 

1/865 

•001156069 

1/744 

•001344086 

1/805 

•001242236 

1/866 

•001154734 

1/745 

•001342282 

1/806 

•001240695 

1/867 

•001153403 

1/746 

•001340483 

1/807 

•001239157 

1/868 

•001152074 

G  2 


78 


THE   PRACTICAL   MODEL   CALCULATOR. 


Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

Fraction  or 
Numb. 

Decimal  or 
Reciprocal. 

1/869 

•001150748 

1/913 

•00109529 

1/957 

•001044932 

1/870 

•001149425 

1/914 

•001094092 

1/958 

•001043841 

1/871 

•001148106 

1/915 

•001092896 

1/959 

•001042753 

1/872 

•001146789 

1/916 

•001091703 

1/960 

•001041667 

1/873 

•001145475 

1/917 

•001090513 

1/961 

•001040583 

1/874 

•001144165 

1/918 

•001089325 

1/962 

•001039501 

1/875 

•001142857 

1/919 

•001088139 

1/963 

•001038422 

1/876 

•001141553 

1/920 

•001086957 

1^964 

•001037344 

1/877 

•001140251 

1/921 

•001085776 

1/965 

•001036269 

1/878 

•001138952 

1/922 

•001084599 

1/966 

•001035197 

1/879 

•001137656 

1/923 

•001083423 

1/967 

•001034126 

1/880 

•001136364 

1/924 

•001082251 

1/968 

•001033058 

1/881 

•001135074 

1/925 

•001081081 

1/969 

•001031992 

1/882 

•001133787 

1/926 

•001079914 

1/970 

•001030928 

1/883 

•001132503 

1/927 

•001078749 

1/971 

•001029866 

1/884 

•001131222 

1/928 

•001077586 

1/972 

•001028807 

1/885 

•001129944 

1/929 

•001076426 

1/973 

•001027749 

1/886 

•001128668 

1/930 

•001075269 

1/974 

•001026694 

1/887 

•001127396 

1/931 

•001074114 

1/975 

•001025641 

1/888 

•001126126 

1/932 

•001072961 

1/976 

•00102459 

1/889 

•001124859 

1/933 

•001071811 

1/977 

•001023541 

1/890 

•001123596 

1/934 

•001070664 

1/978 

•001022495 

1/891 

•001122334 

1/935 

•001069519 

1/979 

•00102145 

1/892 

•001121076 

1/936 

•001068376 

1/980 

•001020408 

1/893 

•001119821 

1/937 

•001067236 

1/981 

•001019168 

1/894 

•001118568 

1/938 

•001066098 

1/982 

•00101833 

1/895 

•001117818 

1/939 

•001064963 

1/983 

•001017294 

1/896 

•001116071 

1/940 

•00106383 

1/984 

•00101626 

1/897 

•001114827 

1/941 

•001062699 

1/985 

•001015228 

1/898 

•001113586 

1/942 

•001061571 

1/986 

•001014199 

1/899 

•001112347 

1/943 

•001060445 

1/987 

•001013171 

1/900 

•001111111 

1/944 

•001059322 

1/988 

•001012146 

1/901 

•001109878 

1/945 

•001058201 

1/989 

•001011122 

1/902 

•001108647 

1/946 

•001057082 

1/990 

•001010101 

1/903 

•00110742 

1/947 

•001055966 

1/991 

•001009082 

1/904 

•001106195 

1/948 

•001054852 

1/992 

•001008065 

1/905 

•001104972 

1/949 

•001053741 

1/993 

•001007049 

1/906 

•001103753 

1/950 

•001052632 

1/994 

•001006036 

1/907 

•001102536 

1/951 

•001051525 

1/995 

•001005025 

1/908 

•001101322 

1/952 

•00105042 

1/996 

•001004016 

1/909 

•00110011 

1/953 

•001049318 

1/997 

•001003009 

1/910 

•001098901 

1/954 

•001048218 

1/998 

•001002004 

1/911 

•001091695 

1/955 

•00104712 

1/999 

•001001001 

1/912 

•001096491 

1/956 

•001046025 

1/1000 

•001 

Divide  80000  by  971. 

By  the  above  Table  we  find  that  1  divided  by  971  gives  -001029866, 
and  -001029866  x  80000  =  82-38928. 
What  is  the  sum  of  -      and 


5  x         =  -001132503  x  5  =  -005662515 


-QQ2098636 


2  X  9M  =  -001049318  x  2 
•'•885  +  958 


07761141 


MENSURATION   OF   SOLIDS. 


79 


WEIGHTS  AND  VALUES  IN  DECIMAL  PARTS. 


TROY  WEIGHT. 

AVOIRDUPOIS 
WEIGHT. 

AVOIRDUPOIS 
WEIGHT. 

Dec.  parts  of  a  Ib. 

Dec.  parts  of  a  cwt. 

Dec.  parts  of  a  Ib. 

Ozs. 

Decimals. 

Qrs. 

Decimals. 

Ozs. 

Decimals. 

11 

•916666 

3 

•75 

15 

•9375 

10 

•833333 

2 

•5 

14 

•875 

9 

•75 

1 

•25 

13 

•8125 

8 

•666666 

Ibs. 

Decimals. 

12 

•75 

7 

•583333 

27 

•241071 

11 

•6875 

6 

•5 

26 

•232142 

10 

•625 

5 

•416666 

25 

•223214 

9 

•5625 

4 

•333333 

24 

•214286 

8 

•5 

3 

•25 

23 

•205357 

7 

•4375 

2 

•166666 

22 

•196428 

6 

•375 

1 

•083333 

21 

•187500 

5 

•3125 

Dwts. 

Decimals. 

20 

•178572 

4 

•25 

19 

•079166 

19 

•169643 

3 

•1875 

18 

•075 

18 

•160714 

2 

•125 

17 

•070833 

17 

•151785 

1 

•0625 

16 

•066666 

16 

•142856 

Drs. 

Decimals. 

15 

•0625 

15 

•133928 

15 

•058593 

14 

•058333 

14 

•125 

14 

•054686 

13 

•054166 

13 

•116071 

13 

•050780 

12 

•05 

12 

•107143 

12 

•046874 

11 

•045833 

11 

•098214 

11 

•042968 

10 

•041666 

10 

•089286 

10 

•039062 

9 

•0375 

9 

•080357 

9 

•035156 

8 

•033333 

8 

•071428 

8 

•03125 

7 

•029166 

7 

•0625 

7 

•027343 

6 

•025 

6 

•053571 

6 

•023437 

5 

•020833 

5 

•044643 

5 

•019531 

4 

•016666 

4 

•035714 

4 

•015625 

3 

•0125 

3 

•026786 

3 

•011718 

2 

•008333 

2 

•017857 

2 

•007812 

1 

•004166 

1 

•008928 

1 

•003906 

Grs. 

Decimals. 

Ozs. 

Decimals. 

15 

14 

•002604 
•002430 

15 
14 

•008370 
•007812 

LONG 
MEASURE. 

13 

•f  O 

•002257 

AAOAOO 

13 

1  O 

•007254 

Dec.  parts  of  a  foot. 

1J 

•UUJUoo 

12 

•006696 

Ins. 

Decimals. 

11 

•001910 

11 

•006138 

11 

•916666 

10 

•001736 

10 

•005580 

10 

•833333 

9 

•001562 

9 

•005022 

9 

•75 

8 

•001389 

8 

•004464 

8 

•666666 

7 

•001215 

7 

•003906 

7 

•583333 

6 

•001042 

6 

•003348 

6 

•5 

5 

•000868 

5 

•002790 

5 

•416666 

4 

•000694 

4 

•002232 

4 

•333333 

3 

•000521 

3 

•001674 

3 

•25 

2 

•000347 

2 

•001116 

2 

•166666 

1 

•000173 

1 

•000558 

1 

•083333 

To  find  the  solidity  of  a  cube,  the  height  of  one 
of  its  sides  being  given. — Multiply  the  side  of  the 
cube  by  itself,  and  that  product  again  by  the  side, 
and  it  will  give  the  solidity  required. 

The  side  AB,  or  BC,  of  the  cube  ABCDFGHE, 
is  25-5  :  what  is  the  solidity  ? 

Here  AB3  =  (22-5)|3  =  25-5  x  25-5  X  25-5  = 
25-5  X  650-25  =  16581-375,  content  of  the  cube. 


II! 


THE  PRACTICAL  MODEL  CALCULATOR. 


To  find  the  solidity  of  a parallelopipedon. 
— Multiply  the  length  by  the  breadth,  and 
that  product  again  by  the  depth  or  altitude, 
and  it  will  give  the  solidity  required. 

Required  the  solidity  of  a  parallelopipedon  G 
ABCDFEHG,  whose  length  AB  is  8  feet, 
its  breadth  FD  4J  feet,  and  the  depth  or 
altitude  AD  6f  feet  ? 

Here  AB  x  AD  X  FD  =  8  X  6-75  x  4.5  =  54  X  4-5  =  243  solid 
feet,  the  contents  of  the  parallelopipedon. 

To  find  the  solidity  of  a  prism. — Multiply  the  area  of  the  base 
into  the  perpendicular  height  of  the  prism,  and  the  product  will  be 
the  solidity. 

What  is  the  solidity  of  the  triangular  prism  ABCF 
ED,  whose  length  AB  is  10  feet,  and  either  of  the 
equal  sides,  BC,  CD,  or  DB,  of  one  of  its  equilateral 
ends  BCD,  2J  feet? 

Here  J  X  2-5*  X  N/3  =  J  X  6-25  X  v/3  =  1-5625 
X  x/3  =  1-5625  X  1-732  ==  2-70625  =  area  of  the 
base  BCD. 

.     2-5  +  2-5  +  2-5       7-5  . 

Or,  —      — g =  -g-  =  3-75  =  J  sum  of 

the  sides,  BC,  CD,  DB,  of  the  triangle  CDB. 

And  3-75  -  2-5  =  1-25, .-.  1-25, 1-25  and  1-25  =  3  differences. 

Whence  N/3-75  X  1-25  X  1-25  X  1-25  =  N/3-75  X  1-253  = 
v/7'32421875  =  2-7063  =  area  of  the  base  as  before, 

And  2-7063  X  10  =  27-063  solid  feet,  the  content  of  the  prism 
required. 

To  find  the  convex  surface  of  a  cylinder. — Multiply  the  peri- 
phery or  circumference  of  the  base,  by  the  height  of  the  cylinder, 
and  the  product  will  be  the  convex  surface. 

D 

What  is  the  convex  surface  of  the  right  cylinder 
ABCD,  whose  length  BC  is  20  feet,  and  the  diame- 
ter of  its  base  AB  2  feet  ? 

Here  3-1416  X  2  =  6-2832  =  periphery  of  the 
base  AB. 

And  6-2832  x  20  =  125-6640  square  feet,  the  A 
convexity  required. 

To  find  the  solidity  of  a  cylinder. — Multiply  the  area  of  the 
base  by  the  perpendicular  height  of  the  cylinder,  and  the  product 
will  be  the  solidity. 

What  is  the  solidity  of  the  cylinder  ABCD,  the  diameter  of 
whose  base  AB  is  30  inches,  and  the  height  BC  50  inches. 

Here  -7854  x  302  =  -7854  X  900  =  706-86  =  area  of  the  base  AB. 

And  706-86  x  50  =  35343  cubic  inches;  or  -,^0-  =  20-4531 


solid  feet. 


1728 


MENSURATION   OF   SOLIDS. 


81 


The  four  following  cases  contain  all  the  rules  for  finding  the  su- 
perficies and  solidities  of  cylindrical  ungulas. 

When  the  section  is  parallel  to  the  axis  of  the  cylinder.  1 
RULE. — Multiply  the  length  of  the  arc  line  of  the  base 

by  the  height  of  the  cylinder,  and  the  product  will  be 

the  curve  surface. 

Multiply  the  area  of  the  base  by  the  height  of  the  A 

cylinder,  and  the  product  will  be  the  solidity. 

When  the  section  passes  obliquely  through  the  opposite 
sides  of  the  cylinder. 

RULE. — Multiply  the  circumference  of  the  base  of  the 
cylinder  by  half  the  sum  of  the  greatest  and  least  lengths 
of  the  ungula,  and  the  product  will  be  the  curve  surface. 

Multiply  the  area  of  the  base  of  the  cylinder  by  half 
the  sum  of  the  greatest  and  least  lengths  of  the  ungula,  and  the 
product  will  be  the  solidity. 

When  the  section  passes  through  the  base  of  the  cylin-  I 
der,  and  one  of  its  sides. 

RULE. — Multiply  the  sine  of  half  the  arc  of  the  base 
by  the  diameter  of  the  cylinder,  and  from  this  product 
subtract  the  product  of  the  arc  and  cosine. 

Multiply  the  difference  thus  found,  by  the  quotient  of  B 
the  height  divided  by  the  versed  sine,  and  the  product 
will  be  the  curve  surface. 

From  f  of  the  cube  of  the  right  sine  of  half  the  arc  of  the  base, 
subtract  the  product  of  the  area  of  the  base  and  the  cosine  of  the 
said  half  arc. 

Multiply  the  difference,  thus  found,  by  the  quotient  arising  from 
the  height  divided  by  the  versed  sine,  and  the  product  will  be  the 
solidity. 

When  the  section  passes  obliquely  through  both  ends 
of  the  cylinder. 

RULE. — Conceive  the  section  to  be  continued,  till  it 
meets  the  side  of  the  cylinder  produced ;  then  say,  as 
the  difference  of  the  versed  sines  of  half  the  arcs  of  the 
two  ends  of  the  ungula  is  to  the  versed  sine  of  half  the 
arc  of  the  less  end,  so  is  the  height  of  the  cylinder  to 
the  part  of  the  side  produced. 

Find  the  surface  of  each  of  the  ungulas,  thus  formed,  and  their 
difference  will  be  the  surface. 

In  like  manner  find  the  solidities  of  each  of  the  ungulas,  and 
their  difference  will  be  the 


To  find  the  convex  surface  of  a  right  cone. — Multiply  the  circum- 
ference of  the  base  by  the  slant  height,  or  the  length  of  the  side 
of  the  cone,  and  half  the  product  will  be  the  surface  required. 

The  diameter  of  the  base  AB  is  3  feet,  and  the  slant  height 
AC  or  BC  15  feet;  required  the  convex  surface  of  the  cone 
ACB. 

6 


82 


THE  PRACTICAL  MODEL  CALCULATOR. 


Here  3-1416  X  3  =  94248  =  circumference  of  the  ba»e  AB. 
9-4248  x  15       141-3720 


, 
Ana 


2 


Tv'Oop  square  feet,  the  convex 


surface  required. 

To  find  the  convex  surface  of  the  frustum  of  a  right  cone. — Mul- 
tiply the  sum  of  the  perimeters  of  the  two  ends,  by  the  slant  height 
of  the  frustum,  and  half  the  product  will  be  the  surface  required. 

c 

In  the  frustum  ABDE,  the  circumferences  of 
the  two  ends  AB  and  DE  are  22-5  and  15-75 
respectively,  and  the  slant  height  BD  is  26 ;  what 
is  the  convex  surface  ? 


Here 


(22-5  +  15-75)  x  26 
~~2  ' 


22-5  +  15-75 


x!3 

face. 


•25  X  13  ==  497-25  =  convex  sur- 


To  find  the  solidity  of  a  cone  or  pyramid. — Multiply  the  area  of 
the  base  by  one-third  of  the  perpendicular  height  of  the  cone  or 
pyramid,  and  the  product  will  be  the  solidity. 


Required  the  solidity  of  the  cone  ACB,  whose 
diameter  AB  is  20,  and  its  perpendicular  height 
OS  24. 

Here  -7854  X  202  =  -7854  X  400  =  314-16 
=  area  of  the  base  AB. 

And  314-16  X  ~  =  314-16  x  8  =  2513-28 
o 

=  solidity  required.  A 

Required  the  solidity  of  the  hexagonal  pyra- 
mid ECBD,  each  of  the  equal  sides  of  its  base 
being  40,  and  the  perpendicular  height  CS  60. 

Here  2*598076  (multiplier  when  the  side  is  1) 
x  402  =  2-598076  x  1600  =  4156-9216  =  area 
of  the  base. 

And  4156-9216  X  ^  =  4156-9216  x  20  = 
83138-432  solidity. 


To  find  the  solidity  of  a  frustum  of  a  cone  or  pyramid. — For 
the  frustum  of  a  cone,  the  diameters  or  circumferences  of  the  two 
ends,  and  the  height  being  given. 

Add  together  the  square  of  the  diameter  of  the  greater  end,  the 
square  of  the  diameter  of  the  less  end,  and  the  product  of  the  two 


MENSURATION   OF   SOLIDS. 


83 


diameters ;  multiply  the  sum  by  -7854,  and  the  product  by  the 
height ;  J  of  the  last  product  will  be  the  solidity.     Or, 

Add  together  the  square  of  the  circumference  of  the  greater 
end,  the  square  of  the  circumference  of  the  less  end,  and  the  pro- 
duct of  the  two  circumferences ;  multiply  the  sum  by  -07958,  and 
the  product  by  the  height ;  J  of  the  last  product  will  be  the  solidity. 

For  the  frustum  of  a  pyramid  whose  sides  are  regular  polygons. — 
Add  together  the  square  of  a  side  of  the  greater  end,  the  square 
of  a  side  of  the  less  end,  and  the  product  of  these  two  sides ;  mul- 
tiply the  sum  by  the  proper  number  in  the  Table  of  Superficies,  and 
the  product  by  the  height ;  J  of  the  last  product  will  be  the  solidity. 

WJien  the  ends  of  the  pyramids  are  not  regular  polygons. — Add 
together  the  areas  of  the  two  ends  and  the  square  root  of  their 
product ;  multiply  the  sum  by  the  height,  and  J  of  the  product 
will  be  the  solidity. 

What  is  the  solidity  of  the  frustum  of  the  cone 
EABD,  the  diameter  of  whose  greater  end  AB  is 
5  feet,  that  of  the  less  end  ED,  3  feet,  and  the 
perpendicular  height  Ss,  9  feet  ? 

(52  +  32  +  5"x~3)  x -7854x9        346-3614 

3  ~~3~ 

115-4538  solid  feet,  the  content  of  the  frustum. 

What  is  the  solidity  of  the  frustum  eEDB6  of  a 
hexagonal  pyramid,  the  side  ED  of  whose  greater 
end  is  4  feet,  that  el  of  the  less  end  3  feet,  and 
the  height  S«,  9  feet  ? 

(42  +  32  +Tx~3)  x  2-598076  x  9       865-159308 
~3~  ~3~ 

=  288-386436  solid  feet,  the  solidity  required. 

The  following  cases  contain  all  the  rules  for  finding  the  superficies 
and  solidities  of  conical  ungulas. 

When  the  section  passes  through  the  opposite  extremities  of  the 
ends  of  the  frustum. 

Let  D  =  AB  the  diameter  of  the  greater  end ; 
d  =  CD,  the  diameter  of  the  less  end ;  h  =  perpen- 
dicular height  of  the  frustum,  and  n  =  *7854. 

d2  —  d  >/Dd      nDh 
Then  — p  _  ,• —  X  — 5—  =  solidity  of  the  greater 

elliptic  ungula  ADB. 

D  \/T)d  —  d2      ndh 

X  -7T-  =  solidity  of  the  less  ungula  ACD. 


D  —  d 


n  h 


D  —  d     X  ~3~  =  Difference  °f  tnese  hoofs. 


surface  of  ADB. 


+  (D  -  c?)  x  (D2  --  3 


=*  curve 


84  THE  PRACTICAL  MODEL  CALCULATOR. 

When  the  section  cuts  off  parts  of  the  base,  and  makes  the  angle 
DrB  less  than  the  angle  CAB. 

Let  S  =  tabular  segment,  whose  versed  sine  is       c 
Br-r-D;  s  —  tab.  seg.  whose  versed  sine  is  Br  —  (D  —  d) 
-7-  d,  and  the  other  letters  as  above. 

Br  Br 

*      "^  /^  ^^      T~\  ~TTV  •»     V^       TT\  TX  ~l       •*».  : 


x  K~f-3  =  solidity  of  the  elliptic  hoof  EFBD. 


And  f: -j  -N/4A2  4-  (D  —  d)2x(seg.  FBE—  f^X- 

.L'  '  " "  Cv  JL/  c&       jt\.y 

X  \/^  _  ^r  X  seg.  of  the  circle  AB,  whose  height  is  D  X  — ^— 
=  convex  surface  of  EFBD. 

When  the  section  is  parallel  to  one  of  the  sides  of  the  frustum. 

Let  A  =  area  of  the  base  FBE,  and  the  other  let-  C(==^.D 
ters  as  before. 


Then 


$  _  d  —  |d  V(B  —  d)  X  d)  X  ±h  =  solidity 


of  the  parabolic  hoof  EFBD. 


And      _     \/4A2  X  (D  -  d)2  X  (seg.  FBE  -  f  D-d! 


X  Vd  X  D  —  d)  =  convex  surface  of  EFBD. 

WJien  the  section  cuts  off  part  of  the  base,  and  makes  the  angle 
DrB  greater  than  the  angle  CAB. 

Let  the  area  of  the  hyperbolic  section  EDF  =  A, 
and  the  area  of  the  circular  seg.  EBF  =  a. 

&h  d  X  Er 

Then  p  _  ^  X  (a  X  D  —  A  X      ^      )  =  solidity  of 

the  hyperbolic  ungula  EFBD. 

And  p^-^  X  v/4A2  +  (D  -  d)2  X  (cir.  seg.  EBF  - 

c—,  x ^  -  v/ —  =  curve  surface  of  EFBD. 


d  x  Cr 
The  transverse  diameter  of  the  hyp.  seg.  =  j>  _  ^  —  gr  anc*  tne 

conjugate  =  c?  \/g  --  d  —  J$9  ^rom  w^c^  ^ts  area  ma^  ^e  ^oun(^ 
by  the  former  rules. 

To  find  the  solidity  of  a  cuneus  or  wedge.  —  Add  twice  the  length 
of  the  base  to  the  length  of  the  edge,  and  reserve  the  number. 

Multiply  the  height  of  the  wedge  by  the  breadth  of  the  base, 
and  this  product  by  the  reserved  number  ;  J  of  the  last  product 
will  be  the  solidity. 


MENSURATION  OF   SOLIDS. 


85 


How  many  solid  feet  are  there  in  a  wedge, 
whose  base  is  5  feet  4  inches  long,  and  9  inches 
broad,  the  length  of  the  edge  being  3  feet  6  inches, 
and  the  perpendicular  height  2  feet  4  inches  ? 


Here 
170  x  28  X  9 


(64  x  2  +  42)  x  28  x  9 

6 
170  x  28  x  3 


C  B 

(128  +  42)  x  28  x  9 
6 

170  x  14  x  3  =  7140  solid 


inches. 

And  7140  -r-  1728  =  4-1319  solid  feet,  the  content. 

To  find  the  solidity  of  a  prismoid. — To  the  sum  of  the  areas  of 
the  two  ends  add  four  times  the  area  of  a  section  parallel  to  and 
equally  distant  from  both  ends,  and  this  last  sum  multiplied  by  \ 
of  the  height  will  give  the  solidity. 

The  length  of  the  middle  rectangle  is  equal  to  half  the  sum  of 
the  lengths  of  the  rectangles  of  the  two  ends,  and  its  breadth  equal 
to  half  the  sum  of  the  breadths  of  those  rectangles. 

What  is  the  solidity  of  a  rectangle  prismoid, 
the  length  and  breadth  of  one  end  being  14  and 
12  inches,  and  the  corresponding  sides  of  the  other 
6  and  4  inches,  and  the  perpendicular  30J  feet. 


Here  14  x  12  +  6  x  4  =  168  +  24  =  192  = 
sum  of  the  area  of  the  two  ends. 

14  +  6      20 
Also  — o —  =  -9-  =  10  =  length  of  the  middle  rectangle. 


And 


12  +  4      16 


breadth  of  the  middle  rectangle. 

4  times  the  area  of 


Whence  10  X  8  X  4  =  80  X  4  =  320 
the  middle  rectangle. 

Or  (320  +  192)  x  -g-  =  512  X  61  =  31232  solid  inches. 

And  31232  -4-  1728  =  18-074  solid  feet,  the  content. 

To  find  the  convex  surface  of  a  sphere. — Multiply  the  diameter 
of  the  sphere  by  its  circumference,  and  the  product  will  be  the 
convex  superficies  required. 

The  curve  surface  of  any  zone  or  segment  will  also  be  found  by 
multiplying  its  height  by  the  whole  circumference  of  the  sphere. 


What  is  the  convex  superficies  of  a  globe 
BOG  whose  diameter  BG  is  17  inches  ? 

Here  3-1416  X  17  X  17  =  53-4072  X  17  = 
907*9224  square  inches. 

And  907-9224  -f-  144  =  6-305  square  feet. 


86 


THE  PRACTICAL  MODEL  CALCULATOR. 


To  find  the  solidity  of  a  sphere  or  globe. — Multiply  the  cube  of 
the  diameter  by  -5236,  and  the  product  will  be  the  solidity. 

What  is  the  solidity  of  the  sphere  AEBC, 
•whose  diameter  AB  is  17  inches  ? 

Here  173  X  -5236  =  17  X  17  X  17  X  -5236  = 


289  x  17  x  5236  =  4913  x 
solid  inches. 
^22572-4468-7-1728  = 


•5236  =  2572-4468 
=  1  -48868  solid  feet 


To  find  the  solidity  of  the  segment  of  a  sphere. — To  three  times 
the  square  of  the  radius  of  its  base  add  the  square  of  its  height,  and 
this  sum  multiplied  by  the  height,  and  the  product  again  by  *5236, 
will  give  the  solidity.  Or, 

From  three  times  the  diameter  of  the  sphere  subtract  twice  the 
height  of  the  segment,  multiply  by  the  square  of  the  height,  and 
that  product  by  -5236  ;  the  last  product  will  be  the  solidity. 

The  radius  Cn  of  the  base  of  the  segment 
CAD  is  7  inches,  and  the  height  An  4  inches ; 
what  is  the  solidity  ? 

Here  (72  x  3  +  42)  x  4  x  -5236  =  (49x3+42) 
x4  x  -5236  =  (147  +  42)  x4  x  -5236  =  (147+16) 
x  4  x  -5236  =  163  x  4  x  -5236  =  652  x  -5236 
==  341-3872  solid  inches. 

To  find  the  solidity  of  a  frustum  or  zone  of  a  sphere. — To  the 
sum  of  the  squares  of  the  radii  of  the  two  ends,  add  one-third  of 
the  square  of  their  distance,  or  of  the  breadth  of  the  zone,  and 
this  sum  multiplied  by  the  said  breadth,  and  the  product  again  by 
1-5708,  will  give  the  solidity. 

What  is  the  solid  content  of  the  zone  ABCD, 
whose  greater  diameter  AB  is  20  inches,  the 
less  diameter  CD  15  inches,  and  the  distance 
nm  of  the  two  ends  10  inches  ? 


Here  (102  +  7'52  -f        )  X  10  X  1-5708  = 

,  (100  +  56-25  +  33-33)  x  10  x  1-5708  =  189-58 
x  10  x  1-5708  =  1895-8  x  1-5708  =  2977-92264  solid  inches. 

To  find  the  solidity  of  a  spheroid.— Multiply  the  square  of  the 
revolving  axe  by  the  fixed  axe,  and  this  product  again  by  -5236, 
and  it  will  give  the  solidity  required. 

•5236  is  =  i  of  3-1416. 

In  the  prolate  spheroid  ABCD,  the 
transverse,  or  fixed  axe  AC  is  90,  and 
the  conjugate  or  revolving  axe  DB  is  70  ; 
what  is  the  solidity  ? 

Here  DB2  x  AC  X  -5236  =  702  X 
x  -5236  =  4900  x  90  x  -5236  =  441000 
X  -5236  =  230907-6  =  solidity  required. 


MENSURATION   OF   SOLIDS. 


8T 


To  find  the  content  of  the  middle  frustum  of  a  splieroid,  its 
length,  the  middle  diameter,  and  that  of  either  of  the  ends,  being 
given,  when  the  ends  are  circular  or  parallel  to  the  revolving  axis. — 
To  twice  the  square  of  the  middle  diameter  add  the  square  of  the 
diameter  of  either  of  the  ends,  and  this  sum  multiplied  by  the  length 
of  the  frustum,  and  the  product  again  by  -2618,  will  give  the  solidity. 

Where  -2618  =  &  of  3-1416. 

In  the  middle  frustum  of  a  spheroid 
EFGH,  the  middle  diameter  DB  is 
50  inches,  and  that  of  either  of  the 
ends  EF  or  GH  is  40  inches,  and  its 
length  nm  18  inches ;  what  is  its  soli- 
dity ? 

Here  (502  X  2  +  402)  X  18  X  -2618 
=  (2500  x  2  +  1600)  x  18  x  -2618  =  (5000  +  1600)  x  18  x 
•2618  =  6600  x  18  X  -2618  =  118800  X  -2613  =  31101-84  cubic 
inches. 

When  the  ends  are  elliptical  or  perpendicular  to  the  revolving 
axis. — Multiply  twice  the  transverse  diameter  of  the  middle  sec- 
tion by  its  conjugate  diameter,  and  to  this  product  add  the  product 
of  the  transverse  and  conjugate  diameters  of  either  of  the  ends. 

Multiply  the  sum  thus  found  by  the  distance  of  the  ends  or 
the  height  of  the  frustum,  and  the  product  again  by  -2618,  and  it 
will  give  the  solidity  required. 

In  the  middle  frustum  ABCD  of  an  oblate  _*_ 

spheroid,  the  diameters  of  the  middle  section 
EF  are  50  and  30,  those  of  the  end  AD  40 
and  24,  and  its  height  ne  18;  what  is  the 
solidity? 

Here  (50  X  2  X  30  +  40  X  24)  X  18  X  -2618 
=  (3000  +  960)  x  18  x  -2618  =  3960  x  18  x 
•2618  =  71280  x  -2618  =  18661-104  =  the  solidity. 

To  find  the  solidity  of  the  segment  of  a  spheroid,  when  the  base 
is  parallel  to  the  revolving  axis. — Divide  the  square  of  the  revolv- 
ing axis  by  the  square  of  the  fixed  axe,  and  multiply  the  quotient 
by  the  difference  between  three  times  the  fixed  axe  and  twice  the 
height  of  the  segment. 

Multiply  the  product  thus  found  by  the  square  of  the  height  of 
the  segment,  and  this  product  again  by  -5236,  and  it  will  give  the 
solidity  required. 

In  the  prolate  spheroid  DEFD,  the  trans- 
verse axis  2  DO  is  100,  the  conjugate  AC  60, 
and  the  height  Dn  of  the  segment  EDF  10 ; 
what  is  the  solidity  ? 

602      

Here  (Tm  x  300  -  20)  X  102  X  -5236  =     . 

A  0~"~  0 

•36  x  280  x  102  x  -5236  =  100-80  x  100  x  -5236  =  10080  x 
•5236  =  5277-888  =  the  solidity. 


88 


THE  PRACTICAL  MODEL  CALCULATOR. 


When  the  base  is  perpendicular  to  the  revolving  axis. — Divide  the 
fixed  axe  by  the  revolving  axe,  and  multiply  the  quotient  by  the 
difference  between  three  times  the  revolving  axe  and  twice  the 
height  of  the  segment. 

Multiply  the  product  thus  found  by  the  square  of  the  height  of 
the  segment,  and  this  product  again  by  -5236,  and  it  will  give  the 
solidity  required. 

In  the  prolate  spheroid  «E6F,  the  trans- 
verse axe  EF  is  100,  the  conjugate  ab  60,  and 
the  -height  an  of  the  segment  aAD  12 ;  what 
is  the  solidity  ? 

Here  156  (=  diff.  of  3ab  and  2an)  X  If 
(=  EF  -f-  ab  X  144  (=  square  of  an)  X  -5236 

156  x  5 
= g x  144  x  -5236  =  52  x  5  x  144  x  -5236  =  260  x 

144  x  -5236  =  37440  x  -5236  =  19603-584  =  the  solidity. 

To  find  the  solidity  of  a  parabolic  conoid. — Multiply  the  area  of 
the  base  by  half  the  altitude,  and  the  product  will  be  the  content. 

What  is  the  solidity  of  the  paraboloid  ADB, 
whose  height  Dm  is  84,  and  the  diameter  BA 
of  its  circular  base  48  ? 

Dm) 


2304  x 

76001-5872 


Here  482  X  -7854  x  42  ( 
•7854  x  42  =  1809-5616  x  42 
=  the  solidity. 

To  find  the  solidity  of  the  frustum  of  a  paraboloid,  when  its  ends 
are  perpendicular  to  the  axe  of  the  solid. — Multiply  the  sum  of  the 
squares  of  the  diameters  of  the  two  ends  by  the  height  of  the  frus- 
tum, and  the  product  again  by  -3927,  and  it  will  give  the  solidity. 

Required  the  solidity  of  the  parabolic  frus- 
tum ABC^,  the  diameter  AB  of  the  greater  end 
being  58,  that  of  the  less  end  do  30,  and  the 
height  no  18. 

Here  (582  +  302)  x  18  x  -3927  =  (3364  -f- 
900)  x  18  x  -3927  =  4264  x  18  x  -3927  = 
76752  X  -3927  =  30140-5104  =  the  solidity. 

To  find  the  solidity  of  an  hyperboloid. — To  the  square  of  the 
radius  of  the  base  add  the  square  of  the  middle  diameter  between 
the  base  and  the  vertex,  and  this  sum  multiplied  by  the  altitude, 
and  the  product  again  by  -5236  will  give  the  solidity. 

In  the  hyperboloid  ACB,  the  altitude  O 
is  10,  the  radius  Ar  of  the  base  12,  and  the  mid- 
dle diameter  nm  15-8745 ;  what  is  the  solidity  ? 

Here  15-87452  +   122   X   10    X    -5236   = 
251-99975  + 144  x  10  x  -5236  =  395-99975  x 
10  x  -5236  =  3959-9975  x  -5236  =  2073-454691  A 
=  the  solidity. 


MENSURATION   OF   SOLIDS. 


89 


To  find  the  solidity  of  the  frustum  of  an  hyperbolic  conoid.  —  Add 
together  the  squares  of  the  greatest  and  least  semi-diameters,  and 
the  square  of  the  whole  diameter  in  the  middle  ;  then  this  sum  being 
multiplied  by  the  altitude,  and  the  product  again  by  '5236,  will 
give  the  solidity. 

In  the  hyperbolic  frustum  ADCB,  the  length 
rs  is  20,  the  diameter  AB  of  the  greater  end  32, 
that  DC  of  the  less  end  24,  and  the  middle  dia- 
meter nm  28-1708;  required  the  solidity. 

Here  (162  +  122  +  28-17082)  X  20  x  -52359 
=  (256  +  144  +  793-5939)  x  20  x  -52359  = 
1193-5939  x  20  x  -52359  =  23871-878  x  -52359 
=  12499-07660202  =  solidity. 

To  find  the  solidity  of  a  tetraedron.  —  Multiply  -f$ 
of  the  cube  of  the  linear  side  by  the  square  root  of 
2,  and  the  product  will  be  the  solidity. 

The  linear  side  of  a  tetraedron  ABCtt  is  4  ;  what 
is  the  solidity  ? 

43         .      4x4x4  _  4x4  16 

Xv/2-—  —  x  v/2- 


16 

=  -7T    X 


12 
1-414 


22-624 


7-5413  =  solidity. 


To  find  the  solidity  of  an  octaedron. — Multiply  J  of  the  cube 
of  the  linear  side  by  the  square  root  of  2,  and  the  product  will  be 
the  solidity. 


"What  is  the  solidity  of  the  octaedron  BGAD, 
whose  linear  side  is  4  ? 


43 


_  64 
—  o   x 


21-333  x  1-414 


V  2  =  21-333,  x  </2 
30-16486  =  solidity. 


To  find  the  solidity  of  a  dodecaedron. — To  21  times  the  square 
root  of  5  add  47,  and  divide  the  sum  by  40 :  then  the  square  root 
of  the  quotient  being  multiplied  by  five  times  the  cube  of  the  linear 
side  will  give  the  solidity. 

The  linear  side  of  the  dodecaedron  ABODE 
is  3 ;  what  is  the  solidity  ? 

x27x5^21x2-2f°6+4T* 


46-95726  +  47 


x  135  =  206-901 


40 

x  27  x  5  = 

solidity. 

To  find  the  solidity  of  an  icosaedron. — To  three  times  the  square 
root  of  5  add  7,  and  divide  the  sum  by  2 :  then  the  square  root  of 

H2 


90 


THE  PRACTICAL  MODEL  CALCULATOR. 


this  quotient  being  multiplied  by  |  of  the  cube  of  the  linear  side 
will  give  the  solidity. 

7  4-  3  v/  5 
That  is  |  S3  X  v/  ( g )  =  solidity  when  S  is  =  to  the 

linear  side. 

The  linear  side  of  the  icosaedron  ABCDEF  D 

is  3  ;  what  is  the  solidity  ? 

Q  .  /  *  _L  »?      5x32 


5  x 

2 
27 

A 
=      <%/ 

6 

6 

•70818 

v 

t 

2 
5 

x 

9 

/V 

6 

2 

2 

B 

=  x/6-85409  x  22-5  =  2-61803 


X  22-5  =  58-9056  = 

The  superficies  and  solidity  of  any  of  the  five  regular  bodies  may 
be  found  as  follows  : 

RULE  1.  Multiply  the  tabular  area  by  the  square  of  the  linear 
edge,  and  the  product  will  be  the  superficies. 

2.  Multiply  the  tabular  solidity  by  the  "cube  of  the  linear  edge, 
and  the  product  will  be  the  solidity. 


Surfaces  and  Solidities  of  the  Regular  Bodies. 


No.  of 
Sides. 

Names. 

Surfaces. 

Solidities. 

4 
6 

8 
A2 

20 
. 

Tetraedron 
Hexaedron 
Octaedron 
Dodecaedron 
Icosaedron 

1.73205 
6.00000 
3.46410 

20.64578 
8.66025 

0.11785 

1.00000 
0.47140 
7.66312 
2.18169 

To  find  the  convex  superficies  of  a  cylindric  ring. — To  the  thick- 
ness of  the  ring  add  the  inner  diameter,  and  this  sum  being  multi- 
plied by  the  thickness,  and  the  product  again  by  9.8696,  will  give 
the  superficies. 

The  thickness  of  Ac  of  a  cylindric  ring  is  3 
inches,  and  the  inner  diameter  cd  12  inches  ; 
what  is  the  convex  superficies  ?  A 

12~+~3  x  3  x  9-8696  =  15  x  3  x  9-8696 
=  45  X  9-8696  =  444-132  =  superficies. 

To  find  the  solidity  of  a  cylindric  ring.— To  the  thickness  of  the 
ring  add  the  inner  diameter,  and  this  sum  being  multiplied  by  the 
square  of  half  the  thickness,  and  the  product  again  by  9-8696, 
will  give  the  solidity. 


MENSURATION   OF   SOLIDS.  91 

What  is  the  solidity  of  an  anchor  ring,  whose  inner  diameter  is 
8  inches,  and  thickness  in  metal  3  inches  ? 

8~T~3  x  f2  x  9-8696  =  11  x  1-52  x  9-8693  =  11  x  2-25  x 
9-8696  =  24-75  X  9-8696  =  244-2726  =  solidity. 

The  inner  diameter  AB  of  the  cylindric  ring 
cdef  equals  18  feet,  and  the  sectional  diameter 
cA.  or  Be  equals  9  inches ;  required  the  convex 
surface  and  solidity  of  the  ring.  

18  feet  X  12  =  216  inches,  and  216  +  9 
x  9  X  9-8696  =  19985-94  square  inches. 

216+  9  x  92  x  2-4674  =  44968-365  cubic 
inches. 

In  the  formation  of  a  hoop  or  ring  of  wrought  iron,  it  is  found 
in  practice  that  in  bending  the  iron,  the  side  or  edge  which  forms 
the  interior  diameter  of  the  hoop  is  upset  or  shortened,  while  at 
the  same  time  the  exterior  diameter  is  drawn  or  lengthened  ;  there- 
fore, the  proper  diameter  by  which  to  determine  the  length  of  the 
iron  in  an  unbent  state,  is  the  distance  from  centre  to  centre  of  the 
iron  of  which  the  hoop  is  composed :  hence  the  rule  to  determine 
the  length  of  the  iron.  If  it  is  the  interior  diameter  of  the  hoop 
that  is  given,  add  the  thickness  of  the  iron  ;  but  if  the  exterior  di- 
ameter, subtract  from  the  given  diameter  the  thickness  of  the  iron, 
multiply  the  sum  or  remainder  by  3-1416,  and  the  product  is  the 
length  of  the  iron,  in  equal  terms  of  unity. 

Supposing  the  interior  diameter  of  a  hoop  to  be  32  inches,  and 
the  thickness  of  the  iron  1 J,  what  must  be  the  proper  length  of  the 
iron,  independent  of  any  allowance  for  shutting  ? 

32  +  1-25  =  33-25  x  3-1416  =  104-458  inches. 
But  the  same  is  obtained  simply  by  inspection  in  the  Table  of  Cir- 
cumferences. 
Thus,  33-25  =  2  feet  9J  in.,  opposite  to  which  is  8  feet  8J  inches. 

Again,  let  it  be  required  to  form  a  hoop  of  iron  J  inch  in  thick- 
ness, and  16J  inches  outside  diameter. 

16-5  -  -875  =  15-625,  or  1  foot  3|  inches ; 
opposite  to  which,  in  the  Table  of  Circumferences,  is  4  feet  1  inch, 
independent  of  any  allowance  for  shutting. 

The  length  for  angle  iron,  of  which  to  form  a  ring  of  a  given  di- 
ameter, varies  according  to  the  strength  of  the  iron  at  the  root ; 
and  the  rule  is,  for  a  ring  with  the  flange  outside,  add  to  its  required 
interior  diameter,  twice  the  extreme  strength  of  the  iron  at  the 
root ;  or,  for  a  ring  with  the  flange  inside,  sub-  c  d  c  d 
tract  twice  the  extreme  strength ;  and  the  sum  or  |/"~'  '"""SI 

remainder  is  the  diameter  by  which  to  determine       TA" ~~%\ 

the  length  of  the  angle  iron.     Thus,  suppose  two       i 

angle  iron  rings  similar  to  the  following  be  re-       L? P.1 

quired,   the  exterior  diameter  AB,  and  interior  ^J\__  __JL=, 

diameter  CD,  each  to  be  1  foot  10J  inches,  and    c  d  c  d 

the  extreme  strength  of  the  iron  at  the  root  cd,  cd,  &c,  {  of  an  inch ; 


92  THE  PRACTICAL  MODEL  CALCULATOR. 

twice  |  =  If,  and  1  ft.  10J  in.  -f  If  =  2  ft.  J  in.,  opposite  to 
whjch,  in  the  Table  of  Circumferences,  is  6  ft.  4J  in.,  the  length  of 
the  iron  for  CD ;  and  1  ft.  10J  in.  —  1}  =  1  ft.  8}  in.,  opposite 
to  which  is  5  ft.  5|  in.,  the  length  of  the  iron  for  AB. 

But  observe,  as  before,  that  the  necessary  allowance  for  shutting 
must  be  added  to  the  length  of  the  iron,  in  addition  to  the  length 
as  expressed  by  the  Table. 

Required  the  capacity  in  gallons  of  a 
locomotive  engine  tender  tank,  2  feet  8 
inches  in  depth,  and  its  superficial  di- 
mensions the  following,  with  reference 
to  the  annexed  plan : 

Length,  or  dist.  between  A  and  B  =  10  ft.  2f  in.  or,  122-75  in. 
Breadth  "  C  and  D  =  6  7j  79-5 

Length  "  i    and  g  =    3       10}  46*75 

Mean  breadth  of  coke- 1   ^  -    3         U  37-25 

space  or  j 

Diameter  of  circle  rn  =    2         8J  32-25 

"  ps  =    1         6J  18-5 

Radius  of  back  corners     vx  =  4  4 

Then,  122-75  X  79-5  =  9758-525  square  inches,  as  a  rectangle. 

And     18-52  X  -7854  =    268-8          "          "       area  of  circle 
formed  by  the  two  ends. 

Total  10027-325  "  "  from  which  de- 
duct the  area  of  the  coke-space,  and  the  difference  of  area  between 
the  semicircle  formed  by  the  two  back  corners,  and  that  of  a  rect- 
angle of  equal  length  and  breadth  ; 

Then  46-75    X  37-25  =  1731-4375  area  of  r,  n,  «,  t,  in  sq.  ins. 

32-252  x  -7854 

— 2 —     — =    408-4        area  of  half  the  circle  rn. 

Radius  of  back  corners  =  4  inches  ; 

consequently  82  X  -7854  =  25-13,  the  semicircle's  area;  and 
8  X  4  =  32  -  25-13  =  6-87  inches  taken  off  by  rounding 
the  corners. 


Hence,  1731-4375  +  408-4  +  6-87  =  2146-707,  and 

10027-235  -  2146-707  =  7880-618  square  inches,  or 

whole  area  in  plan, 

7880-618  x  32  the  depth  =  252179-776  cubic  inches, 
and  252179-776  divided  by  231  gives  1091-6873  the 
content  in  gallons. 


MENSURATION   OF   TIMBER. 


93 


TABLES  ly  which  to  facilitate  the  Mensuration  of  Timber. 
1.  Flat  or  Board  Measure. 


Breadth  in 
inches. 

Area  of  a 
liiieal  foot. 

Breadth  in 
inches. 

Area  of  a 
lineal  foot. 

Breadth  in 
inches. 

Area  of  a 
lineal  foot. 

I 

•0208 

4 

•3334 

8 

•6667 

f 

•0417 

44 

•3542 

^ 

•6875 

2. 

•0625 

4 

•375 

8* 

•7084 

1* 

•0834 

4 

•3958 

8| 

•7292 

U 

•1042 

5 

•4167 

9 

•75 

l| 

•125 

5£ 

•4375 

9£ 

•7708 

If 

•1459 

01 

•4583 

9} 

•7917 

2 

•1667 

5f 

•4792 

9| 

•8125 

2| 

•1875 

•6 

•5 

10 

•8334 

2* 

•2084 

6i 

•5208 

lof 

•8542 

2f 

•2292 

01 

•5416 

10} 

•875 

3 

•25 

6| 

•5625 

10| 

•8959 

31 

•2708 

7 

•5833 

11 

•9167 

3* 

•2916 

7i 

•6042 

1H 

•9375 

3f 

•3125 

4 

•625 

ill 

•9583 

7| 

•6458 

iif 

•9792 

Application  and  Use  of  the  Table. 

Required  the  number  of  square  feet  in  a  board  or  plank  16 J  feet 
in  length  and  9f  inches  in  breadth. 

Opposite  9|  is  -8125  X  16-5  =  13-4  square  feet. 
A  board  1  foot  2f  inches  in  breadth,  and  21  feet  in  length ;  what 
is  its  superficial  content  in  square  feet  ? 

Opposite  2f  is  -2292,  to  which  add  the  1  foot ;  then 
1-2292  x  21  =  25-8  square  feet. 

In  a  board  15  J  inches  at  one  end,  9  inches  at  the  other,  and 
14 J  feet  in  length,  how  many  square  feet  ? 

^-  -  =  12J,  or  1-0208;  and  1-0208  X  14-5  =  14-8  sq.  ft. 

The  solidity  of  round  or  unsquared  timber  may  be  found  with 
much  more  accuracy  by  the  succeeding  Rule  : — Multiply  the  square 
of  one-fifth  of  the  mean  girth  by  twice  the  length,  and  the  product 
will  be  the  solidity,  very  near  the  truth. 

A  piece  of  timber  is  30  feet  long,  and  the  mean  girth  is  128  in- 
ches, what  is  the  solidity  ? 

128 


Then 


25-6 


5 
x  60 


144 


273-06  cubic  feet. 


This  is  nearer  the  truth  than  if  one-fourth  the  girth  be  em- 
ployed. 


THE  PRACTICAL  MODEL  CALCULATOR. 


2.  Cubic  or  Solid  Measure. 


Mean  X 
girt  in 
inches. 

Cubic  feet 
in  each 
lineal  foot. 

Mean  % 
girt  in 
inches. 

Cubic  feet 
in  each 
lineal  foot. 

Mean  % 
girt  in 
inches. 

Cubic  feet 
in  each 
lineal  foot. 

Mean  >^ 
girt  in 
inches. 

Cubic  feet 
in  each 
lineal  foot. 

6 

•25 

12 

1 

18 

2-25 

24 

4 

6J 

•272 

w 

1-042 

18i 

2-313 

241 

4-084 

6£ 

•294 

12f 

1-085 

18} 

2-376 

24-£ 

4-108 

6| 

•317 

12| 

1-129 

18| 

2-442 

24-| 

4-254 

7 

•340 

13 

1-174 

19 

2-506 

25 

4-34 

7| 

•364 

131 

1-219 

m 

2-574 

251 

4-428 

•39 

18} 

1-265 

19} 

2-64 

26} 

4-516 

7f 

•417 

13| 

1-313 

19f 

2-709 

25J 

4-605 

8 

•444 

14 

1-361 

20 

2-777 

26 

4-694 

8J 

•472 

141 

1-41 

20-1 

2-898 

261 

4-785 

8* 

•501 

14} 

1-46 

20£ 

2-917 

26} 

4-876 

8} 

•531 

14| 

1-511 

20* 

2-99 

26f 

4-969 

9 

•562 

15 

1-562 

21 

3-062 

27 

5-062 

9i 

•594 

151 

1-615 

211 

3-136 

271 

5-158 

9| 

•626 

15} 

1-668 

21£ 

3-209 

27£ 

5-252 

9f 

•659 

16f 

1-772 

2lf 

3-285 

27f 

5-348 

10 

•694 

16 

1-777 

22 

3-362 

28 

5-444 

101 

•73 

16J 

1-833 

221 

3-438 

281 

5-542 

10J 

•766 

16} 

1-89 

22} 

3-516 

28} 

5-64 

lOf 

•803 

16| 

1-948 

22f 

3-598 

28| 

5-74 

11 

•84 

17 

2-006 

23 

3-673 

29 

5-84 

11| 

•878 

17i 

2-066 

23J 

3-754 

291 

5-941 

Hi 

•918 

17} 

2-126 

23£ 

3-835 

29} 

6-044 

ill 

•959 

17f 

2-187 

28} 

3-917 

29| 

6-146 

In  the  cubic  estimation  of  timber,  custom  has  established  the 
rule  of  J,  the  mean  girt  being  the  side  of  the  square  considered  as 
the  cross  sectional  dimensions ;  hence,  multiply  the  number  of  cubic 
feet  by  lineal  foot  as  in  the  Table  of  Cubic  Measure  opposite  the 
J  girt,  and  the  product  is  the  solidity  of  the  given  dimensions  in 
cubic  feet. 

Suppose  the  mean  J  girt  of  a  tree  21J  inches,  and  its  length 
16  feet,  what  are  its  contents  in  cubic  feet  ? 

3-136  x  16  =  50-176  cubic  feet. 

Battens,  Deals,  and  Planks  are  each  similar  in  their  various 
lengths,  but  differing  in  their  widths  and  thicknesses,  arid  hence 
their  principal  distinction :  thus,  a  batten  is  7  inches  by  2 J,  a  deal 
9  by  3,  and  a  plank  11  by  3,  these  being  what  are  termed  the 
standard  dimensions,  by  which  they  are  bought  and  sold,  the  length 
of  each  being  taken  at  12  feet ;  therefore,  in  estimating  for  the 
proper  value  of  any  quantity,  nothing  more  is  required  than  their 
lineal  dimensions,  by  which  to  ascertain  the  number  of  times  12  feet, 
there  are  in  the  given  whole. 

Suppose  I  wish  to  purchase  the  following : 


7  of 
5 

11 
and    6 


6  feet 
14 
19 
21 


6  x 
14  x 
19  x 
21  x 


7  = 

5  = 
11  = 

6  - 


42  feet 

70 
209 
126 


12)447)  37'25  standard  deals. 


MENSURATION   OF   TIMBER. 


95 


TABLE  showing  the  number  of  Lineal  Feet  of  Scantling  of  various 
dimensions,  which  are  equal  to  a  Cubic  Foot. 


& 

03 
CD 

A 
O 

.2 

<N 

Inches. 

4 

1 

5 
<» 

f 
i 

Ft.    In. 

4  inches  by 

Inches. 

j 

1 

.3 

e 

I 

Ft.       In. 

* 

03 

s 

,2 

6 

g 

o 

Inches. 

| 

S 

.a 
1 

a1 
a> 
ft 

Ft.     In. 

2 

? 

t} 

45* 
5} 
6 

«» 

? 
? 

1? 

10, 

"i 

12 

36    0 
28    9 
24    0 
20    7 
18    0 
16    0 
14    5 
13    1 
12     0 
11     1 
10     3 
9    7 
9    0 
8    6 
8    0 
7    7 
7    3 
6  10 
6     6 
6    4 
6    0 

4 

f 

^ 
6 

•i 

? 

8* 
9 

»l 
10 
10} 

11 

m 

12 

9     0 
8     0 
7     2 
6     6 
6     0 
5     6 
5     1 
4     9 
4     6 
4     3 
4     0 
3     9 
3     7 
3     6 
3     3 
3     2 
3     0 

9£ 
10 
10J 

11 

s* 

2      G 
2     5 
2     3 
2     2 
2      1 
2     0 

& 

S 

-= 

.2 
t~ 

7 

? 

S* 

1? 

10, 

11} 

12 

2    11 
2     9 
2     6 
2     5 
2     3 
2     2 
2     1 
1    11 
1    10 
1      9 
1      8 

£ 

• 

a> 

1 

iO 

5 

? 

?} 
I* 

i1 

9£ 
10 
10, 

i^ 

5     9 
5     3 
4   10 
4     5 
'4     1 
3    10 
3     7 
3     5 
3     2 
3     0 
2    10 
2      9 
2     8 
2      6 
2     4 

£ 

1 

o 

.s 

CO 

8 
? 

1? 
10, 

"* 

12 

2      3 
2      1 

2     0 
1    10 
1      9 
1     8 
1      7 
1     7 
1      6 

£ 

t» 
| 

o 

.2 

03 

3 

f 

*} 

5 

? 

«» 

? 
? 

i? 

101 

11J 

12 

16    0 
13     8 
12    0 
10    8 
9    7 
9    0 
8    0 
7    4 
6  10 
6    4 
6    0 
5    8 
5    4 
5    0 
4  10 
4    6 
4    4 
4    2 
4    0 

* 

1 

os 

9 

lo1* 

10, 

MI 

12 

1     9 
1     8 
1     7 
1     6 
1     5 
1     4 
1     4 

£ 

1 
w 

.2 

co 

6 

? 
? 

8J 
9 

4      0 
3      8 
3      5 
3     2 
3      0 
2    10 
2      8 

£ 

.2 
o 

T—l 

10 
10, 

"i 

1     5 
1      4 
1      4 
1      3 

Hewn  and  sawed  timber  are  measured  by  the  cubic  foot.  The 
unit  of  board  measure  is  a  superficial  foot  one  inch  thick. 

To  measure  round  timber. — Multiply  the  length  in  feet  by  the 
square  of  J  of  the  mean  girth  in  inches,  and  the  product  divided 
by  144  gives  the  content  in  cubic  feet. 

The  J  girths  of  a  piece  of  timber,  taken  at  five  points,  equally 
distant  from  each  other,  are  24,  28,  33,  35,  and  40  inches ;  the 
length  30  feet,  what  is  the  content  ? 

24  +  28  +  33  +  35  +  40 

=  32. 


Then 


5 
x  30 


213J  cubic  feet. 


96 


THE   PRACTICAL   MODEL   CALCULATOR. 


TABLE  containing  the  Superficies  and  Solid  Content  of  Spheres,  from 
1  to  12,  and  advancing  by  a  tenth. 


Diam. 

Superficies. 

Solidity. 

Diam. 

Superficies. 

Solidity. 

Diam. 

Superficies. 

Solidity. 

1-0 

3-1416 

•5236 

4-7 

69-3979 

54-3617 

8-4 

221-6712 

310-3398 

•1 

3-8013 

•6969 

•8 

72-3824 

57-9059 

•5 

226-9806 

321-5558 

•2 

4-5239 

•9047 

•9 

75-4298 

61-6010 

•6 

232-3527 

333-0389 

•3 

5-3093 

1-1503 

5-0 

78-5400 

65-4500 

•7 

237-7877 

344-7921 

•4 

6-1575 

1-4367 

•1 

81-7130 

69-4560 

•8 

243-2855 

356-8187 

•5 

7-0686 

1-7671 

.2 

84-9488 

73-6223 

•9 

248-8461 

369-1217 

•6 

8-0424 

2-1446 

•3 

88-2475 

77-9519 

9-0 

254-4696 

381-7044 

•7 

9-0792 

2-5724 

•4 

91-6090 

82-4481 

•1 

260-1558 

394-5697 

•8 

10-1787 

3-0536 

•5 

95-0334 

87-1139 

•2 

265-9130 

407-7210 

•9 

11-3411 

3-5913 

•6 

98-5205 

91-9525 

•3 

271-7169 

421-1613 

2-0 

12-5664 

4-1888 

•7 

102-0705 

96-9670 

•4 

277-5917 

434-8937 

•1 

13-8544 

4-8490 

•8 

105-6834 

102-1606 

•5 

283-5294 

448-9215 

•2 

15-2053 

5-5752 

•9 

109-3590 

107-5364 

•6 

289-5298 

463-2477 

•3 

16-6190 

63706 

6-0 

113-0976 

113-0976 

•7 

295-5931 

477-7755 

•4 

18-0956 

7-2382 

-,  -1 

116-8989 

118-8472 

•8 

301-7192 

492-8081 

•5 

19-6350 

8-1812 

-2 

120-7631 

124-7885 

•9 

307-9082 

508-0485 

•6 

21-2372 

9-2027 

•3 

124-6901 

130-9246 

10-0 

314-1600 

523-6000 

•7 

22-9022 

10-3060 

•4 

128-6799 

137-2585 

•1 

320-4746 

539-4656 

•8 

24-6300 

11-4940 

•5 

132-7326 

143-7936 

•2 

326-8520 

555-6485 

•9 

26-4208 

12-7700 

•6 

136-8480 

150-5329 

•3 

333-2923 

572-1518 

3.0 

28-2744 

14-1372 

•7 

141-0264 

157-4795 

•4 

339-7954 

588-9784 

•1 

30-1907 

15-5985 

•8 

145-2675 

164-6365 

•5 

346-3614 

606-1324 

•2 

32-1699 

17-1573 

•9 

149-5715 

172-0073 

•6 

352-9901 

623-6159 

•3 

34-2120 

18-8166 

7-0 

153-9384 

179-5948 

•7 

359-6817 

641-4325 

•4 

36-3168 

20-5795 

•1 

158-3680 

187-4021 

•8 

366-4362 

659-5852 

•5 

38-4846 

22-4493 

•2 

162-8605 

195-4326 

•9 

373-2534 

678-0771 

•6 

40-7151 

24-4290 

•3 

167-4158 

203-6893 

11-0 

380-1336 

696-9116 

•7 

43-0085' 

26-5219 

•4 

172-0340 

212-1752 

•1 

387-0765 

716-0915 

•8 

45-3647 

28-7309 

•5 

176-7150 

220-8937 

•2 

394-0823 

735-6200 

•9 

47-7837 

31-0594 

•6 

181-4588 

229-8478 

•3 

401-1509 

755-5008 

4-0 

50-2656 

33-5104 

•7 

186-2654 

239-0511 

•4 

408-2823 

775-7364 

•1 

52-8102 

36-0870 

•8 

191-1349 

248-4754 

•5 

415-4766 

796-3301 

•2 

55-4178 

38-7924 

•9 

196-0672 

258-1552 

•6 

422-7336 

817-2851 

•3 

58-0881 

41-6298 

8-0 

201-0624 

268-0832 

•7 

430-0536 

838-6045 

•4 

60-8213 

44-6023 

•1 

206-1203 

278-2625 

•8 

437-4363 

860-2915 

•5 

63-6174 

47-7130 

•2 

211-2411 

288-6962 

•9 

444-8819 

882-3492 

•6 

66-4782 

50-9651 

•3 

216-4248 

299-3876 

12-0 

452-3904 

904-7808 

To  reduce  Solid  Inches  into  Solid  Feet. 


1728  Solid  Inches  to  one  Solid  Foot. 

Feet.  Inches. 

Feet.      Inches. 

Feet.    Inches. 

Feet.      Inches. 

Feet.      Inches. 

Feet.         Inches. 

1  =  1728 

18=31104 

35=60480 

52=88956 

69  =  119232 

85=146880 

2     3456 

19     32832 

36     62208 

53     91584 

70     120960 

86     148608 

3     5184 

20     34560 

37     63936 

54     93312 

71     122688 

87     150336 

4     6912 

21     36288 

38     65664 

55     95040 

72     124416 

88     152064 

5     8640 

22     38016 

39     67392 

56     96768 

73     126144 

89     153792 

6   10368 

23     39744 

40     69120 

57     98496 

74     127872 

90     155520 

7    12096 

24     41472 

41      70848 

58   100224 

75     129600 

91      157248 

8   13824 

25     43200 

42     72576 

59   101952 

76     131328 

92     158976 

9   15552 

26     44928 

43     74304 

60   103680 

77     133056 

93     160704 

10   17280 

27     46656 

44     76032 

61   105408 

78     134784 

94     162432 

11   19008 

28     48384 

45     77760 

62   107136 

79     136512 

95     164160 

12   20736 

29     50112 

46     79488 

63   108864 

80     138240 

96     165888 

13  22464 

30     51840 

47      81216 

64  110592 

81      139968 

97     167616 

14  24192 

31     53568 

48     82944 

65   112320 

82     141696 

98     169344 

15  25920 

32     55296 

49     84672 

66  114048 

83     143424 

99     171072 

16  27648 

33     57024 

50     86400 

67   115776 

84     145152 

100     172800 

17   29376 

34     58752 

51     88128 

68  117504 

CUTTINGS   AND    EMBANKMENTS. 


9T 


CUTTINGS  AND  EMBANKMENTS. 

THE  angle  of  repose  upon  railways,  or  that  incline  on  which  a 
carriage  would  rest  in  whatever  situation  it  was  placed,  is  said  to 
be  at  1  in  280,  or  nearly  19  feet  per  mile ;  at  any  greater  rise 
than  this,  the  force  of  gravity  overcomes  the  horizontal  traction, 
and  carriages  will  not  rest,  or  remain  quiescent  upon  the  line,  but 
will  of  themselves  run  down  the  line  with  accelerated  velocity. 
The  angle  of  practical  effect  is  variously  stated,  ranging  from  1  in 
75  to  1  in  330. 

The  width  of  land  required  for  a  railway  must  vary  with  the 
depth  of  the  cuttings  and  length  of  embankments,  together  with 
the  slopes  necessary  to  be  given  to  suit  the  various  materials  of 
which  the  cuttings  are  composed :  thus,  rock  will  generally  stand 
when  the  sides  are  vertical ;  chalk  varies  from  ^  to  1,  to  1  to  1 ; 
gravel  1J  to  1 ;  coal  1 J  to  1 ;  clay  1  to  1,  &c. ;  but  where  land 
can  be  obtained  at  a  reasonable  rate,  it  is  always  well  to  be  on  the 
safe  side. 

The  following  Table  is  calculated  for  the  purpose  of  ascertain- 
ing the  extent  of  any  cutting  in  cubic  yards,  for  1  chain,  22  yards, 
or  66  feet  in  length,  the  slopes  or  angles  of  the  sides  being  those 
which  are  most  in  general  practice,  and  formation  level  equal  30  feet. 

Slopes  1  to  1. 


Depth 
of 
cut- 
ting in 
feet. 

Half 

width 
at 
top  in 
feet. 

Content 
iu  cubic 
yards  per 
chain. 

Content 
of  1  per- 
jendicu- 
,ar  ft.  in 
breadth. 

Content 
of  3  per- 
pendicu- 
lar  ft.  in 
breadth. 

Content 
of  6  per- 
pendicu- 
lar ft.  in 
breadth. 

Depth 
of 
cut- 
ting in 
feet. 

Half 

width 
at 
top  in 
feet. 

Content 
in  cubic 
yards  per 
chain. 

Content 
of  1  per- 
pendicu- 
lar ft.  in 
breadth. 

Content 
of  3  per- 
pendicu- 
lar ft.  in 
breadth. 

Content 
of  6  per- 
pendicu- 
lar ft.  in 
breadth. 

1 

16 

75-78 

2-44 

7-33 

14-67 

26 

41 

3599-11 

63-55 

190-67 

381-33 

2 

17 

156-42 

4-89 

14-67 

29-33 

27 

42 

3762-00 

65-99 

198-00 

396-00 

3 

18 

242-00 

7-33 

22-00 

44-00 

28 

43 

3969-78 

68-43 

205-33 

410-67 

4 

19 

332-44 

9-78 

29-33 

58-67 

29 

44 

4182-44 

70-88 

212-67 

425-33 

5 

20 

427-78 

12-22 

36-67 

73-33 

30 

45 

4400-00 

73-32 

220-00 

440-00 

6 

21 

528-00 

14-67 

44-00 

88-00 

31 

46 

4622-44 

75-77 

227-33 

454-67 

7 

22 

633-11 

17-11 

51-33 

102-67 

32 

47 

4849-78 

78-22 

234-67 

469-33 

8 

23 

743-11 

19-56 

58-67 

117-33 

33 

48 

5082-00 

80-67 

242-00 

484-00 

9 

24 

858-00 

22-00 

66-00 

132-00 

34 

49 

5319-11 

83-11 

249-33 

498-67 

10 

25 

977-78 

24-44 

73-33 

146-67 

35 

50 

5561-11 

85-55 

256-67 

513-33 

11 

26 

1102-44 

26-89 

80-67 

161-33 

36 

51 

5808-00 

88-00 

264-00 

528-00 

12 

27 

1232-00 

29-33 

88-00 

176-00 

37 

52 

6059-78 

90-44 

271-33542-67 

13 

28 

1366-44 

31-78 

95-33 

190-67 

38 

53 

6316-44 

92-39 

278-67 

557-33 

14 

29 

1505-78 

34-22 

102-67 

205-33 

39 

54 

6578-00 

95-33 

286-00 

572-00 

15 

30 

1650-00 

36-66 

110-00 

220-00 

40 

55 

6844-44 

97-77 

293-33 

586-67 

16 

31 

1799-11 

39-11 

117-33 

234-67 

41 

56 

7115-78 

100-22 

300-67 

601-33 

17 

32 

1953-11 

41-55 

124-67 

249-33 

42 

57 

7392-00 

102-66 

308-00 

616-00 

18 

33 

2112-00 

43-99 

132-00 

264-00 

43 

58 

7673-11 

105-11 

315-33 

630-67 

19 

34 

2275-78 

46-44 

139-33 

278-67 

44 

59 

7959-11 

107-55 

322-67 

645-33 

20 

35 

2444-44 

48-89 

146-67 

293-33 

45 

60 

8250-00 

109-99 

330-00!660-00 

21 

36 

2618-00 

51-33 

154-00 

308-00 

46 

61 

8545-78 

112-44 

337-33674-67 

22 

37 

2796-44 

53-77 

161-33 

322-67 

47 

62 

8846-44 

114-88 

344-67689-33 

23 

38 

2979-78 

56-21 

168-67 

337-33 

48 

63 

9152-00 

117-33 

352-00704-00 

24 

39 

3168-00 

58-66 

176-00 

352-00 

49 

64 

9462-44 

119-77 

359-33718-67 

25 

40 

3361-11 

61-10 

183-33 

366-67 

50 

65 

9777-78 

122-21 

366-67 

733-33 

\  B  R  A 

OF   THK 

TTKfT V  KRSITT 


98 


THE  PRACTICAL  MODEL  CALCULATOR. 


Slopes  l 


!  Depth 

Half 

Content 

Content 

Content 

Depth 

Half 

Content 

Content 

Content 

of 

cut- 

width 
at 

Content 
in  cubic 

of  1  per- 
pendicu- 

of 3  per- 
pendicu- 

of 6  per- 
pendicu- 

of 
cat- 

width 
at 

Content 
ia  cubic 

of  1  per- 
pendicu- 

of 3  per- 
pendicu- 

of 6  per- 
pendicu- 

ting iu 

top  in 

yards  per 

lar  ft.  in 

lar  ft.  in 

lar  ft.  in 

ting  in 

top  in 

yards  per 

lar  ft.  in 

lar  ft.  in 

lar  ft.  ia 

;     ied. 

feet. 

chain. 

breadth. 

breadth. 

breadth. 

feet. 

feet. 

chain. 

breadth. 

breadth. 

breadth. 

1 

16i 

77-00 

2-44 

7-33 

14-67 

26 

54 

4385-33 

63-55 

190-67 

381-33 

2 

18 

161-33 

4-89 

14-67 

29-33 

27 

55| 

4653-00 

65-99 

198-00 

396-00 

3 

19* 

253-00 

7-33 

22  '00 

44-00 

28 

57 

4928-00 

68-43 

205-33 

410-67 

4 

21 

352-00 

9-78 

29-33 

58-67 

29 

58i 

5210-33 

70-88 

212-67 

425-33 

5 

22J 

453-33 

12-22 

36-67 

73-33 

30 

60 

5500-00 

73-32 

220-00 

440-00 

6 

24 

572-00 

14-67 

44-00 

88-00 

31 

611 

5797-00 

75-77 

227-33 

454-67 

7 

25* 

693-00 

17-11 

51-33 

102-67 

32 

63 

6101-33 

78-22 

234-67 

469-33 

8 

27" 

821-33 

19-56 

58-67 

117-33 

33 

641 

6413-00 

80-67 

242-00 

484-00 

9 

28* 

957-00 

22-00 

66-00 

132-00 

34 

66 

6732-00 

83-11 

249-33 

498-67 

10 

30 

1100-00 

24-44 

73-33 

146-67 

35 

6<£ 

7058-33 

85-55 

256-67 

513-33 

11 

31* 

1250-33 

26-89 

80-67 

161-33 

36 

69 

7392-00 

88-00 

264-00 

528-00 

12 

33 

1408-00 

29-33 

88-00 

176-00 

37 

70i 

7733-00 

90-44271-33 

542-67 

13 

34J 

1573-00 

31-78 

95-33 

190-67 

38 

72 

8081-33 

92-39 

278-67 

557-33 

14 

36 

1745-33 

34-22 

102-67 

205-33 

39 

731 

8437-00 

95-33 

286-00 

572-00 

15 

87* 

1925-00 

36-66 

110-00 

220-00 

40 

75 

8800-00 

97-77 

293-33 

586-67 

16 

39 

2112-00 

39-11 

117-33 

234-67 

41 

76* 

9170-33 

100-22 

300-67 

601-33 

17 

40i 

2306-33 

41-55 

124-67 

249-33 

42 

78 

9548-00 

102-66 

308-00 

616-00 

18 

42 

2508-00 

43-99 

132-00 

264-00 

43 

79-1 

9933-00 

105-11 

315-33 

630-67 

19 

43i 

2717-00 

46-44 

139-33 

278-67 

44 

81 

10325-33 

107-55 

322-67 

645-33 

20 

45 

2933-33 

48-89 

146-67 

293-33! 

45 

82* 

10725-00 

109-99J  330-00 

660-00 

21 

46J 

3157-00 

51-33 

154-00 

308-00; 

46 

84 

11132-00 

112-44  337-33 

674-67 

22 

48 

3388-00 

53-77 

161-33 

322-67 

47 

85* 

11546-33 

114-88;  344-67 

689-33 

23 

49i 

3626-33 

56-21 

168-67 

337-331 

48 

87 

11968-00 

117-33  352-00 

704-00 

24 

51 

3872-00 

58-66 

176-00 

352-00 

49 

88* 

12397-00 

119-77 

359-33 

718-67 

25 

62* 

4125-00 

61-10 

183-33 

866-67 

50 

90 

12833-33 

122-21 

366-67 

733-33 

Slopes  2  to  1. 


Depth 
of 
cut- 
ting la 

feet. 

Half 
width 
at 
top  in 
feet. 

Content 
in  cubic 
yards  per 
chain. 

Content 
of  1  per- 
nendicu- 
lar  ft.  in 
breadth. 

Content 
of  3  per- 
pendicu- 
lar ft.  in 
breadth. 

Content 
of  6  per- 
>endicu- 
ar  ft.  in 
Breadth. 

Depth 
of 
cut- 
ting in 
feet. 

Half 

width 
at 
top  in 

Content 
in  cubic 
yards  per 
chain. 

Content 
of  1  per- 
pendicu- 
lar ft.  ii 
breadth. 

Content 
of  3  per- 
pendicu- 
lar ft.  in 
breadth. 

Content 
of  6  per- 
pendicu- 
lar ft.  in 
breadth. 

1 

17 

78-22 

2-44 

7-33 

14-67 

26 

67 

5211-55 

63-55 

190-67 

381-33 

2 

19 

166-22 

4-89 

14-67 

29-33 

27 

69 

5544-00 

65-99 

198-00 

396-00 

3 

21 

264-00 

7-33 

22-00 

44-00 

28 

71 

5886-22 

68-43 

205-33 

410-67 

4 

23 

371-55 

9-78 

29-33 

58-67 

29 

73 

6238-22 

70-88 

212-67 

425-33 

5 

25 

488-89 

12-22 

36-67 

73-33 

30 

75 

6600-00 

73-32 

220-00 

440-00 

6 

27 

616-00 

14-67 

44-00 

88-00 

31 

77 

6971-55 

75-77 

227-33 

454-67 

7 

29 

752-89 

17-11 

51-33 

102-67 

32 

79 

7352-89 

78-22 

234-67 

469-33 

8 

31 

899-55 

19-56 

58-67 

17-33 

33 

81 

7744-00 

80-67 

242-00 

484-00 

9 

33 

1056-00 

22-00 

66-00 

32-00 

34 

83 

8144-89 

83-11 

249-33 

498-67 

10 

35 

1222-22 

24-44 

73-33 

46-67 

35 

85 

8555-55 

85-55 

256-67 

513-33 

11 

37 

1398-22 

26-89 

80-67 

61-33 

36 

87 

8976-00 

88-00 

264-00 

528-00 

12 

39 

1584-00 

29-33 

88-00 

76-00 

37 

89 

9406-22 

90-44 

271-33 

542-67 

13 

41 

1779-55 

31-78 

95-33 

9067 

38 

91 

9846-22 

92-39 

278-67 

557-33 

14 

43 

1984-89 

34-22 

102-67 

05-33 

39 

93 

10296-00 

95-33 

286-00 

572-00 

15 

45 

2200-00 

36-66 

110-00 

20-00 

40 

95 

10755-55 

97-77 

293-33  586-67 

16 

47  2424-89 

39-11 

117-33 

34-67 

41 

97 

11224-89 

100-22 

300-67 

601-33 

17 

49  2659-55 

41-55 

124-67 

49-33 

42 

99 

11704-00 

102-66 

308-00 

616-00 

18 

51  2904-00 

43-99 

132-00 

64-00 

43 

101 

12192-89 

105-11 

315-33 

630-67 

19 

53  3158-22 

46-44 

139-33 

78-67 

44 

103 

12691-55 

107-55 

322-67 

645-33 

20 

55  34-2222 

48-89 

146-67 

93-33 

45 

105 

13200-00 

109-99 

330-00 

660-00 

21 

57 

3696-00 

51-33 

154-00 

08-00 

46 

107 

13718-22 

112-44 

337-33 

674-67 

22 

59 

3979-55 

53-77 

161-33 

22-67 

47 

109 

14246-221114-88 

344-67  689-33 

23 

61  4272-89 

56-21 

168-67 

37-33 

48 

111 

14784-00 

117-33 

352-00 

704-00 

24 

63  J4576-00 

58-66 

176-00 

52-00 

49 

113 

15331-55 

119-77 

359-33 

718-67 

25 

65  4888-89 

61-10 

183-33 

66-67 

50 

115 

15888-89 

122-21 

366-67 

733-33 

CUTTINGS   AND   EMBANKMENTS.  99 

By  the  fourth,  fifth,  and  sixth  columns  in  each  table,  the  number 
of  cubic  yards  is  easily  ascertained  at  any  other  width  of  formation 
level  above  or  below  30  feet,  having  the  same  slopes  as  by  the 
tables,  thus  :  — 

Suppose  an  excavation  of  40  feet  in  depth,  and  33  feet  in  width 
at  formation  level,  whose  slopes  or  sides  are  at  an  angle  of  2  to  1, 
required  the  extent  of  excavation  in  cubic  yards  : 

10755-55  +  293-33  =  11048-88  cubic  yards. 

The  number  of  cubic  yards  in  any  other  excavation  may  be  as- 
certained by  the  following  simple  rule  : 

To  the  width  at  formation  level  in  feet,  add  the  horizontal  length 
of  the  side  of  the  triangle  formed  by  the  slope,  multiply  the  sum 
by  the  depth  of  the  cutting,  or  excavation,  and  by  the  length,  also 
in  feet  ;  divide  the  product  by  27,  and  the  quotient  is  the  content 
in  cubic  yards. 

Suppose  a  cutting  of  any  length,  and  of  which  take  1  chain,  its 
depth  being  14J  feet,  width  at  the  bottom  28  feet,  and  whose  sides 
have  a  slope  of  1  J  to  1,  required  the  content  in  cubic  yards  : 

14-5  x  1-25  =  18-125  -f  28  x  14  =  645-75  x  66  = 

42619*5 
—  ^=  --  =  1578-5  cubic  yards. 

+  rV)  V  +  (b  +  rh)  li  +  4  [i  +  r^^  —  ~ 

gives  the  content  of  any  cutting.  In  words,  this  formula  will  be  :  — 
To  the  area  of  each  end,  add  four  times  the  middle  area  ;  the  sum 
multiplied  by  the  length  and  divided  by  6  gives  the  content.  The 
breadth  at  the  bottom  of  cutting  =  b  ;  the  perpendicular  depth  of 
cutting  at  the  higher  end  =  h  ;  the  perpendicular  depths  of  cutting 
at  the  lower  end  =  Ji'  ;  ?,  the  length  of  the  solid  ;  and  rh'  the  ratio 
of  the  perpendicular  height  of  the  slope  to  the  horizontal  base,  mul- 
tiplied by  the  height  hf.  rh,  the  ratio  r,  of  the  perpendicular 
height  of  the  slope,  to  the  horizontal  base,  multiplied  by  the 
height  h. 

Let  b  =  30  ;  h  =  50  ;  Jif  =  20  ;  I  =  84  feet  ;  and  2  to  5  or  f 
the  ratio  of  the  perpendicular  height  of  the  slope  to  the  horizontal 
base  : 


(30  +  |  x  20)  20  +  (30  +  f  x  50)  50  +  4  [30  +  f  --] 

®~^    \  =  14  |  38  x  20  +  50  x  50  +  4  x  44  x  35  |  =  131880 

131880 
cubic  feet.      —  ^  —  =  4884-44  cubic  yards. 

This  rule  is  one  of  ihe  most  useful  in  the  mensuration  of  solids, 
it  will  give  the  content  of  any  irregular  solid  very  nearly,  whether 
it  be  bounded  by  right  lines  or  not. 


100 


THE   PRACTICAL   MODEL   CALCULATOR. 


TABLE  of  Squares,  Cubes,  Square  and  Cube  Roots  of  Numbers. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1 

1 

1 

1-0000000 

1-0000000 

•100000000 

2 

4 

8 

1-4142136 

1-2599210 

•500000000 

3 

9 

27 

1-7320508 

1-4422496 

•333333333 

4 

16 

64 

2-0000000 

1-5874011 

•250000000 

5 

25 

125 

2-2360680 

1-7099759 

•200000000 

6 

36 

216 

2-4494897 

1-8171206 

•166666667 

7 

49 

343 

2-6457513 

1-9129312 

•142857143 

8 

64 

512 

2-8284271 

2-0000000 

•125000000 

9 

81 

729 

3-0000000 

2-0800837 

•111111111 

10 

100 

1000 

3-1622777 

2-1544347 

•100000000 

11 

121 

1331 

3-3166248 

2-2239801 

•090909091 

12 

144 

1728 

3-4641016 

2-2894286 

•083333333 

13 

169 

2197 

3-6055513 

2-3513347 

•076923077 

14 

196 

2744 

3-7416574 

2-4101422 

•071428571 

15 

225 

3375 

3-8729833 

2-4662121 

•066666667 

16 

256 

4096 

4-0000000 

2-5198421 

•062500000 

17 

289 

4913 

4-1231056 

2-5712816 

•058823529 

18 

324 

5832 

4-2426407 

2-6207414 

•055555556 

19 

361 

6859 

4-3588989 

2-6684016 

•052631579 

20 

400 

8000 

4-4721360 

2-7144177 

•050000000 

21 

441 

9261 

4-5825757 

2-7589243 

•047619048 

22 

484 

10648 

4-6904158 

2-8020393 

•045454545 

23 

529 

12167 

4-7958315 

2-8438670 

•043478261 

24 

576 

13824 

4-8989795 

2-8844991 

•041666667 

25 

625 

15625 

5-0000000 

2-9240177 

•040000000 

26 

676 

17576 

5-0990195 

2-9624960 

•038461538 

27 

729 

19683 

5-1961524 

3-0000000 

•037037037 

28 

784 

21952 

5-2915026 

3-0365889 

•035714286 

29 

841 

24389 

5-3851648 

3-0723168 

•034482759 

30 

900 

27000 

5-4772256 

3-1072325 

•033333333 

31 

961 

29791 

5-5677644 

3-1413806 

•032258065 

32 

1024 

32768 

5-6568542 

3-1748021 

•031250000 

33 

1089 

35937 

5-7445626 

3-2075343 

•030303030 

34 

1156 

39304 

5-8309519 

3-2396118 

•029411765 

35 

1225 

42875 

5-9160798 

3-2710663 

•028571429 

36 

1296 

46656 

6-0000000 

3-3019272 

•027777778 

37 

1369 

50653 

6-0827625 

3-3322218 

•027027027 

38 

1444 

54872 

6-1644140 

3-3619754 

•026315789 

39 

1521 

59319 

6-2449980 

3-3912114 

•025641026 

40 

1600 

64000 

6-3245553 

3-4199519 

•025000000 

41 

1681 

68921 

6-4031242 

3-4482172 

•024390244 

42 

1764 

74088 

6-4807407 

3-4760266 

•023809524 

43 

1849 

79507 

6-5574385 

3-5033981 

•023255814 

44 

1936 

85184 

6-6332496 

3-5303483 

•022727273 

45 

2025 

91125 

6-7082039 

3-5568933 

•022222222 

46 

2116 

97336 

6-7823300 

3-5830479 

•021739130 

47 

2209 

103823 

6-8556546 

3-6088261 

•021276600 

48 

2304 

110592 

6-9282032 

3-6342411 

•020833333 

49 

2401 

117649 

7-0000000 

3-6593057 

•020408163 

50 

2500 

125000 

7-0710678 

3-6840314 

•020000000 

51 

2601 

132651 

7-1414284 

3-7084298 

•019607843 

52 

2704 

140608 

7-2111026 

3-7325111 

•019230769 

53 

2809 

148877 

7-2801099 

3-7562858 

•018867925 

54 

2916 

157464 

7-3484692 

3-7797631 

•018518519 

55 

3025 

166375 

7-4161985 

3-8029525 

•018181818 

56 

3136 

175616 

7-4833148 

3-8258624 

•017857143 

57 

3249 

185193 

7-5498344 

3-8485011 

•017543860 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.         101 


Number. 

Squares, 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

58 

3364 

195112 

7-6157731 

3-8708766 

•017241379 

59 

3481 

205379 

7-6811457 

3-8929965 

•016949153 

60 

3600 

216000 

7-7459667 

3-9148676 

•016666667 

61 

3721 

226981 

7-8102497 

3-9304972 

•016393443 

62 

3844 

238328 

7-8740079 

3-9578915 

•016129032 

63 

3969 

250047 

7-9372539 

3-9790571 

•015873016 

64 

4096 

262144 

8-0000000 

4-0000000 

•015625000 

65 

4225 

274625 

8-0622577 

4-0207256 

•015384615 

66 

4356 

287496 

8-1240384 

4-0412401 

•015151515 

67 

4489 

300763 

8-1853528 

4-0615480 

•014925373 

68 

4624 

314432 

8-2462113 

4-0816551 

•014705882 

69 

4761 

328509 

8-3066239 

4-1015661 

•014492754 

70 

4900 

343000 

8-3666003 

4-1212853 

•014285714 

71 

5041 

357911 

8-4261498 

4-1408178 

•014084517 

72 

5184 

373248 

8-4852814 

4-1601676 

•013888889 

73 

5329 

'  389017 

8-5440037 

4-1793390 

•013698630 

74 

5476 

405224 

8-6023253 

4-1983364 

•013513514 

75 

5625 

421875 

8-6602540 

4-2171633 

•013333333 

76 

5776 

438976 

8-7177979 

4-2358236 

•013157895 

77 

5929 

456533 

8-7749644 

4-2543210 

•012987013 

78 

6084 

474552 

8-8317609 

4-2726586 

•012820513 

79 

6241 

493039 

8-8881944 

4-2908404 

•012658228 

80 

6400 

512000 

8-9442719 

4-3088695 

•012500000 

81 

6561 

531441 

9-0000000 

4-3267487 

.•012345679 

82 

6724 

551368 

9-0553851 

4-3444815 

•012195122 

83 

6889 

571787 

9-1104336 

4-3620707 

•012048193 

84 

7056 

592704 

9-1651514 

4-3795191 

•011904762 

85 

7225 

614125 

9-2195445 

4-3968296 

•011764706 

86 

7396 

636056 

9-2736185 

4-4140049 

•011627907 

87 

7569 

658503 

9-3273791 

4-4310476 

•011494253 

88 

7744 

681472 

9-3808315 

4-4470692 

•011363636 

89 

7921 

704969 

9-4339811 

4-4647451 

•011235955 

90 

8100 

729000 

9-4868330 

4-4814047 

•011111111 

91 

8281 

753571 

9-5393920 

4-4979414 

•010989011 

92 

8464 

778688 

9-5916630 

4-5143574 

•010869565 

93 

8649 

804357 

9-6436508 

4-5306549 

•010752688 

94 

8836 

830584 

9-6953597 

4-5468359 

•010638298 

95 

9025 

857374 

9-7467943 

4-5629026 

•010526316 

96 

9216 

884736 

9-7979590 

4-5788570 

•010416667 

97 

9409 

912673 

9-8488578 

4-5947009 

•010309278 

98 

9604 

941192 

9-8994949 

4-6104363 

•010204082 

99 

9801 

970299 

9-9498744 

4-6260650 

•010101010 

100 

10000 

1000000 

10-0000000 

4-6415888 

•010000000 

101 

10201 

1030301 

10-0498756 

4-6570095 

•009900990 

102 

10404 

1061208 

10-0995049 

4-6723287 

•009803922 

103 

10609 

1092727 

10-1488916 

4-6875482 

•009708738 

104 

10816 

1124864 

10-1980390 

4-7026694 

•009615385 

105 

11025 

1157625 

10-2469508 

4-7176940 

•009523810 

106 

11236 

1191016 

10-2956301 

4-7326235 

•009433962 

107 

11449 

1225043 

10-3440804 

4-7474594 

•009345794 

108 

11664 

1259712 

10-3923048 

4-7622032 

•009259259 

109 

11881 

1295029 

10-4403065 

4-7768562 

•009174312 

110 

12100 

1331000 

10-4880885 

4-7914199 

•009090909 

111 

12321 

1367631 

10-5356538 

4-8058995 

•009009009 

112 

12544 

1404928 

10-5830052 

4-8202845 

•008928571 

113 

12769 

1442897 

10-6301458 

4-8345881 

•008849558 

114 

12996 

1481544 

10-6770783 

4-8488076 

•008771930 

115 

13225 

1520875 

10-7238053 

4-8629442 

•008695652 

116 

13456 

1560896 

10-7703296 

4-8769990 

•008020690 

117 

13689 

1601613 

10-8166538 

4-8909732 

•008547009 

118 

13924 

1643032 

10-8627805 

4-9048681 

•008474576 

119 

14161 

1685159 

10-9087121 

4-9186847 

•008403361 

102 


THE   PRACTICAL   MODEL    CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

120 

14400 

1728000 

10-9544512 

4-9324242 

•008333333 

121 

14641 

1771561 

11-0000000 

4-9460874 

•008264463 

122 

14834 

1815848 

11-0453610 

4-9596757 

•008196721 

123 

15129 

1860867 

11-0905365 

4-9731898 

•008130081 

124 

15376 

1906624 

11-1355287 

4-9866310 

•008064516 

125 

15625 

1953125 

11-1803399 

5-0000000 

•008000000 

126 

15876 

2000376 

11-2249722 

5-0132979 

•007936508 

127 

16129 

2048383 

11-2694277 

5-0265257 

•007874016 

128 

16384 

2097152 

11-3137085 

5-0396842 

•007812500 

129 

16641 

2146689 

11-3578167 

5-0527743 

•007751938 

130 

16900 

2197000 

11-4017543 

5-0657970 

•007692308 

131 

17161 

2248091 

11-4455231 

5-0787531 

•007633588 

132 

17424 

2299968 

11-4891253 

5-0916434 

•007575758 

133 

17689 

2352637 

11-5325626 

5-1044687 

•007518797 

134 

17956 

2406104 

11-5758369 

5-1172299 

•007462687 

135 

18225 

2460375 

11-6189500 

5-1299278 

•007407407 

136 

18496 

2515456 

11-6619038 

5-1425632 

•007352941 

137 

18769 

2571353 

11-7046999 

5-1551367 

•007299270 

138 

19044 

2628072 

11-7473444 

5-1676493 

•007246377 

139 

19321 

2685619 

11-7898261 

5-1801015 

•007194245 

140 

19600 

2744000 

11-8321596 

.5-1924941 

•007142857 

141 

19881 

2803221 

11-8743421 

5-2048279 

•007092199 

142 

20164 

2863288 

11-9163753 

5-2171034 

•007042254 

143 

20449 

2924207 

11-9582607 

5-2293215 

•006993007 

144 

20736 

2985984 

12-0000000 

5-2414828 

•006944444 

145 

21025 

3048625 

12-0415946 

5-2535879 

•006896552 

146 

21316 

3112136 

12-0830460 

5-2656374 

•006849315 

147 

21609 

3176523 

12-1243557 

5-2776321 

•006802721 

148 

21904 

3241792 

12-1655251 

5-2895725 

•006756757 

149 

22201 

3307949 

12-2065556 

5-3014592 

•006711409 

150 

22500 

3375000 

12-2474487 

5-3132928 

•006666667 

151 

22801 

3442951 

12-2882057 

5-3250740 

•006622517 

152 

23104 

3511008 

12-3288280 

5-3368033 

•006578947 

153 

23409 

3581577 

12-3693169 

5-3484812 

•006535948 

154 

23716 

3652264 

12-4096736 

5-3601084 

•006493506 

155 

24025 

3723875 

12-4498996 

5-3716854 

•006451613 

156 

24336 

3796416 

12-4899960 

5-3832126 

•006410256 

157 

24649 

3869893 

12-5299641 

5-3946907 

•006369427 

158 

24964 

3944312 

12-5698051 

5-4061202 

•006329114 

159 

25281 

4019679 

12-6095202 

5-4175015 

•006289308 

160 

25600 

4096000 

12-6491106 

5-4288352 

•006250000 

161 

25921 

4173281 

12-6885775 

5-4401218 

•006211180 

162 

26244 

4251528 

12-7279221 

5-4513618 

•006172840 

163 

26569 

4330747 

12-7671453 

5-4625556 

•006134969 

164 

26896 

4410944 

12-8062485 

5-4737037 

•006097561 

165 

27225 

4492125 

12-8452326 

5-4848066 

•006060606 

166 

27556 

4574296 

12-8840987 

5-4958647 

•006024096 

167 

27889 

4657463 

12-9228480 

5-5068784 

•005988024 

168 

28224 

4741632 

12-9614814 

5-5178484 

•005952381 

169 

28561 

4826809 

13-0000000 

5-5287748 

•005917160 

170 

28900 

4913000 

13-0384048 

5-5396583 

•005882353 

171 

29241 

5000211 

13-0766968 

5-5504991 

•005847953 

172 

29584 

5088448 

13-1148770 

5-5612978 

•005813953 

173 

29929 

5177717 

13-1529464 

5-5720546 

•005780347 

174 

30276 

5268024 

13-1909060 

5-5827702 

•005747126 

175 

30625 

5359375 

13-2287566 

5-5934447 

•005714286 

176 

30976 

5451776 

13-2664992 

5-6040787 

•005681818 

177 

31329 

5545233 

13-3041347 

5-6146724 

•005649718 

178 

31684 

5639752 

13-3416641 

5-6252263 

•005617978 

179 

32041 

6735339 

13-3790882 

5-6357408 

•005586592 

180     32400 

5832000 

13-4164079 

5-6462162 

•005555556 

181     32761 

5929741 

13-4536240 

5-6566528 

•005524862 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.         103 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

182 

33124 

6028568 

13-4907376 

5-6670511 

•005494505 

183 

33489 

6128487 

13-5277493 

5-6774114 

•005464481 

184 

33856 

6229504 

13-5646600 

5-6877340 

•005434783 

185 

34225 

6331625 

13-6014705 

5-6980192 

•005405405 

186 

34596 

6434856 

13-6381817 

5-7082675 

•005376344 

187 

34969 

6539203 

13-6747943 

5-7184791 

•005347594 

188 

35344 

6644672 

13-7113092 

5-7286543 

•005319149 

189 

35721 

6751269 

13-7477271 

5-7387936 

•005291005 

190 

36100 

6859000 

13-7840488 

5-7488971 

•005263158 

191 

36481 

6967871 

13-8202750 

5-7589652 

•005235602 

192 

36864 

7077888 

13-8564065 

5-7689982 

•005208333 

193 

37249 

7189517 

13-8924400 

5-7789966 

•005181347 

194 

37636 

7301384 

13-9283883 

5-7889604 

•005154639 

195 

38025 

7414875 

13-9642400 

5-7988900 

•005128205 

196 

38416 

7529536 

14-0000000 

5-8087857 

•005102041 

197 

38809 

7645373 

14-0356688 

5-8186479 

•005076142 

198 

39204 

7762392 

14-0712473 

5-8284867 

•005050505 

199 

39601 

7880599 

14-1067360 

5-8382725 

•005025126 

200 

40000 

8000000 

14-1421356 

5-8480355 

•005000000 

201 

40401 

8120601 

14-1774469 

5-8577660 

•004975124 

202 

40804 

8242408 

14-2126704 

5-8674673 

•004950495 

203 

41209 

8365427 

14-2478068 

5-8771307 

•004926108 

204 

41616 

8489664 

14-2828569 

5-8867653 

•004901961 

205 

42025 

8615125 

14-3178211 

5-8963685 

•004878049 

2U6 

42436 

8741816 

14-3527001 

5-9059406 

•004854369 

207 

42849 

8869743 

14-3874946 

5-9154817 

•004830918 

208 

43264 

8998912 

14-4222051 

5-9249921 

•004807692 

209 

43681 

9129329 

14-4568323 

5-9344721 

•004784689 

210 

44100 

9261000 

14-4913767 

5-9439220 

•004761905 

211 

44521 

9393931 

14-5258390 

5-9533418 

•004739336 

212 

44944 

9528128 

14-5602198 

5-9627320 

•004716981 

213 

45369 

9663597 

14-5945195 

5-9720926 

•004694836 

214 

45796 

9800344 

14-6287388 

5-9814240 

•004672897 

215 

46225 

9938375 

14-6628783 

5-9907264 

•004651163 

216 

46656 

10077696 

14-6969385 

6-0000000 

•004629630 

217 

47089 

10218313 

14-7309199 

6-0092450 

•004G08295 

218 

47524 

10360232 

14-7648231 

6-0184617 

•004587156 

219 

47961 

10503459 

14-7986486 

6-0276502 

•004566210 

220 

48400 

10648000 

14-8323970 

6-0368107 

•004545455 

221 

48841 

10793861 

14-8660687 

6-0459435 

•004524887 

222 

49284 

10941048 

14-8996644 

6-0550489 

•004504505 

223 

49729 

11089567 

14-9331845 

6-0641270 

•004484305 

224 

50176 

11239424 

14-9666295 

6-0731779 

•004464286 

225 

50625 

11390625 

15-0000000 

6-0824020 

•004444444 

226 

51076 

11543176 

15-0332964 

6-0991994 

•004424779 

227 

51529 

11697083 

15-0665192 

6-1001702 

•004405280 

228 

51984 

11852352 

15-0996689 

6-1091147 

•004385965 

229 

52441 

12008989 

15-1327460 

6-1180332 

•004366812 

230 

52900 

12167000 

15-1657509 

6-1269257 

•004347826 

231 

53361 

12326391 

15-1986842 

6-1357924 

•004329004 

232 

53824 

1248*168 

15-2315462 

6-1446337 

•004310345 

233 

54289 

12649337 

15-2643375 

•  6-1534495 

•004291845 

234 

54756 

12812904 

15-2970585 

6-1622401 

•004273504 

235 

55225 

12977875 

15-3297097 

6-1710058 

•004255319 

236 

55696 

13144256 

15-3622915 

6-1797466 

•004237288 

237 

56169 

13312053 

15-3948043 

6-1884628 

•004219409 

238 

56644 

13481272 

15-4272486 

6-1971544 

•004201681 

239 

57121 

13651919 

15-4596248 

6-2058218 

•004184100 

240 

57600 

13824000 

15-4919334 

6-2144650 

•004166667 

241 

58081 

13997521 

15-5241747 

6-2230843 

•004149378 

242 

58564 

14172488 

15-5563492 

6-2316797 

•004132231 

243 

59049 

14348907 

15-5884573 

6-2402515 

•004115226 

104 


THE    PRACTICAL   MODEL   CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

244 

59536 

14526784 

15-6204994 

6-2487998 

•004098361 

245 

60025 

14706125 

15-6524758 

6-2573248 

•004081633 

246 

60516 

14886936 

15-6843871 

6-2658266 

•004065041 

247 

61009 

15069223 

15-7162336 

6-2743054 

•004048583 

248 

61504 

15252992 

15-7480157 

6-2827613 

•004032258 

249 

62001 

15438249 

15-7797338 

6-2911946 

•004016064 

250 

62500 

15625000 

15-8113883 

6-2996053 

•004000000 

251 

63001 

15813251 

15-8429795 

6-3079935 

•003984064 

252 

63504 

16003008 

15-8745079 

6-3163596 

•003968254 

253 

64009 

16194277 

15-9059737 

6-3247035 

•003952569 

254 

64516 

16387064 

15-9373775 

6-3330256 

•003937008 

255 

65025 

16581375 

15-9687194 

6-3413257 

•003921569 

256 

65536 

16777216 

16-0000000 

6-3496042 

•003906250 

257 

66049 

16974593 

16-0312195 

6-3578611 

•003891051 

258 

66564 

17173512 

16-0623784 

6-3660968 

•003875969 

259 

67081 

17373979 

16-0934769 

6-3743111 

•003861004 

260 

67600 

17576000 

16-1245155 

6-3825043 

•003846154 

261 

68121 

17779581 

16-1554944 

6-3906765 

•003831418 

262 

68644 

17984728 

16-1864141 

6-3988279 

•003816794 

263 

69169 

18191447 

16-2172747 

6-4069585 

•003802281 

264 

69696 

18399744 

16-2480768 

6-4150687 

•003787879 

265 

70225 

18609625 

16-2788206 

6-4231583 

•003773585 

266 

70756 

18821096 

16-3095064 

6-4312276 

•003759398 

267 

71289 

19034163 

16-3401346 

6-4392767 

•003745318 

268 

71824 

19248832 

16-3707055 

6-4473057 

•003731343 

269 

72361 

19465109 

16-4012195 

6-4553148 

•003717472 

270 

72900 

19683000 

164316767 

6-4633041 

•003703704 

271 

73441 

19902511 

16-4620776 

6-4712736 

•003690037 

272 

73984 

20123643 

16-4924225 

6-4792236 

•003676471 

273 

74529 

20346417 

16-5227116 

6-4871541 

•003663004 

274 

75076 

20570824 

16-5529454 

6-4950653 

•003649635 

275 

75625 

20796875 

16-5831240 

6-5029572 

•003636364 

276 

76176 

21024576 

16-6132477 

6-5108300 

•003623188 

277 

76729 

21253933 

16-6433170 

6-5186839 

•003610108 

278 

77284 

21484952 

16-6783320 

6-5265189 

•003597122 

279 

77841 

21717639 

16-7032931 

6-5343351 

•003584229 

280 

78400 

21952000 

16-7332005 

6-5421326 

•003571429 

281 

78961 

22188041 

16-7630546 

6-5499116 

•003558719 

282 

79524 

22425768 

16-7928556 

6-5576722 

•003546099 

283 

80089 

22665187 

16-8226038 

6-5654144 

•003533569 

284 

80656 

22906304 

16-8522995 

6-5731385 

•003522127 

285 

81225 

23149125 

16-8819430 

6-5808443 

•003508772 

286 

81796 

23393656 

16-9115345 

6-5885323 

•003496503 

287 

82369 

23639903 

16-9410743 

6-5962023 

•003484321 

288 

82944 

23887872 

16-9705627 

6-6038545 

•003472222 

289 

83521 

24137569 

17-0000000 

6-6114890 

•003460208 

290 

84100 

24389000 

17-0293864 

6-6191060 

•003448276 

291 

84681 

24642171 

17-0587221 

6-6267054 

•003436426 

292 

85264 

24897088 

17-0880075 

6-6342874 

•003424658 

293 

85849 

25153757 

17-1172428 

6-6418522 

•003412969 

294 

86436 

25412184 

17-1464282 

6-6493998 

•003401361 

295 

87025 

25672375 

17-1755640 

6-6569302 

•003389831 

296 

87616 

25934836 

17-2046505 

6-6644437 

•003378378 

297 

88209 

26198073 

17-2336879 

6-6719403 

•003367003 

298 

88804 

26463592 

-  17-2626765 

6-6794200 

•003355705 

299 

89401 

26730899 

17-2916165 

6-6868831 

•003344482 

300 

90000 

27000000 

17-3205081 

6-6943295 

•003333333 

301 

90601 

27270901 

17-3493516 

6-7017593 

•003322259 

302 

91204 

27543608 

17-3781472 

6-7091729 

•003311258 

303 

91809 

27818127 

17-4068952 

6-7165700 

•003301330 

304 

92416 

28094464 

17-4355958 

6-7239508 

•003289474 

305 

93025 

28372625 

17-4642492 

6-7313155 

•003278689 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.          105 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Eoots. 

Reciprocals. 

306 

93636 

28652616 

17-4928557 

6-7386641 

•003267974 

307 

94249 

28934443 

17-5214155 

6-7459967 

•003257329 

308 

94864 

29218112 

17-5499288 

6-7533134 

•003246753 

309 

95481 

29503609 

17-5783958 

6-7606143 

•003236246 

310 

96100 

29791000 

17-6068169 

6-7678995 

•003225806 

311 

96721 

30080231 

17-6351921 

6-7751690 

•003215434 

312 

97344 

30371328 

17-6635217 

6-7824229 

•003205128 

313 

97969 

30664297 

17-6918060 

6-7896613 

•003194888 

314 

98596 

30959144 

17-7200451 

6-7968844 

•003184713 

'  315 

99225 

31255875 

17-7482393 

6-8040921 

•003174603 

316 

99856 

31554496 

17-7763888 

6-8112847 

•003164557 

817 

100489 

31855013 

17-8044938 

6-8184620 

•003154574 

318 

101124 

32157432 

17-8325545 

6-8256242 

•003144654 

319 

101761 

32461759 

17-8605711 

6-8327714 

•003134796 

320 

102400 

32768000 

17-8885438 

6-8399037 

•003125000 

321 

103041 

33076161 

17-9164729 

6-8470213 

•003115265 

322 

103684 

33386248 

17-9443584 

6-8541240 

•003105590 

323 

104329 

33698267 

17-9722008 

6-8612120 

•003095975 

324 

104976 

34012224 

18-0000000 

6-8682855 

•003086420 

325 

105625 

34328125 

18-0277564 

6-8753433 

•003076923 

326 

106276 

34645976 

18-0554701 

6-8823888 

•003067485 

327 

106929 

34965783 

18-0831413 

6-8894188 

•003058104 

328 

107584 

35287552 

18-1107703 

6-8964345 

•003048780 

329 

108241 

35611289 

18-1383571 

6-9034359 

•003039514 

330 

108900 

35937000 

18-1659021 

6-9104232 

•003030303 

331 

109561 

36264691 

18-1934054 

6-9173964 

•003021148 

332 

110224 

36594368 

18-2208672 

6-9243556 

•003012048 

333 

110889 

36926037 

•   18-2482876 

6-9313088 

•003003003 

334 

111556 

37259704 

18-2756669 

6-9382321 

•002994012 

335 

112225 

37595375 

18-3030052 

6-9451496 

•002985075 

336 

112896 

37933056 

18-3303028 

6-9520533 

•002976190 

337 

113569 

38272753 

18-3575598 

6-9589434 

•002967359 

338 

114244 

38614472 

18-3847763 

6-9658198 

•002958580 

339 

114921 

38958219 

18-4119526 

6-9726826 

•002949853 

340 

115600 

39304000 

18-4390889 

6-9795321 

•002941176 

341 

116281 

39651821 

18-4661853 

6-9863681 

•002932551 

342 

116964 

40001688 

18-4932420 

6-9931906 

•002923977 

343 

117649 

40353607 

18-5202592 

7-0000000 

•002915452 

344 

118336 

40707584 

18-5472370 

7-0067962 

•002906977 

345 

119025 

41063625 

18-5741756 

7-0135791 

•002898551 

346 

119716 

41421736 

18-6010752 

7-0203490 

•002890173 

347 

120409 

41781923 

18-6279360 

7-0271058 

•002881844 

348 

121104 

42144192 

18-6547581 

7-0338497 

•002873563 

349 

121801 

42508549 

18-6815417 

7-0405860 

•002865330 

350 

122500 

42875000 

18-7082869 

7-0472987 

•002857143 

351 

123201 

43243551 

18-7349940 

7-0540041 

•002849003 

352 

123904 

43614208 

18-7616630 

7-0606967 

•002840909 

353 

124609 

43986977 

18-7882942 

7-0673767 

•002832861 

354 

125316 

44361864 

18-8148877 

7-0740440 

•002824859 

355 

126025 

44738875 

18-8414437 

7-0806988 

•002816901 

356 

126736 

45118016 

18-8679623 

7-0873411 

•002808989 

357 

127449 

45499293 

18-8944436 

7-0939709 

•002801120 

358 

128164 

45882712 

18-9208879 

7-1005885 

•002793296 

359 

128881 

46268279 

18-9472953 

7-1071937 

•002785515 

360 

129600 

46656000 

18-9736660 

7-1137866 

•002777778 

361 

130321 

47045831 

19-0000000 

7-1203674 

•002770083 

362 

131044 

47437928 

19-0262976 

7-1269360 

•002762431 

363 

131769 

47832147 

19-0525589 

7-1334925 

•002754821 

364 

132496 

48228544 

19-0787840 

7-1400370 

•002747253 

365 

133225 

48627125 

19-1049732 

7-1465695 

•002739726 

366 

133956 

49027896 

19-1311265 

7-1530901 

•002732240 

367 

134689 

49430863 

19-1572441 

7-1595988 

•002724796 

106 


THE    PRACTICAL   MODEL    CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

368 

135424 

49836032 

19-1833261 

7-1660957 

•002717391 

369 

136161 

50243409 

19-2093727 

7-1725809 

•002710027 

370 

136900 

50653000 

19-2353841 

7-1790544 

•002702703 

371 

137641 

51064811 

19-2613603 

7-1855162 

•002695418 

372 

138384 

51478848 

19-2873015 

7-1919663 

•002688172 

373 

139129 

51895117 

19-3132079 

7-1984050 

•002680965 

374 

139876 

52313624 

19-3390796 

7-2048322 

•002673797 

375 

140625 

52734375 

19-3649167 

7-2112479 

•002666667 

376 

141376 

53157376 

19-3907194 

7-2176522 

•002659574 

377 

142129 

53582633 

19-4164878 

7-2240450 

•002652520 

378 

142884 

54010152 

19-4422221 

7-2304268 

•002645503 

379 

143641 

54439939 

19-4679223 

7-2367972 

•002638521 

380 

144400 

54872000 

19-4935887 

7-2431565 

•002631579 

381 

145161 

55306341 

19-5192213 

7-2495045 

•002624672 

382 

145924 

55742968 

19-5448203 

7-2558415 

•002617801 

383 

146689 

56181887 

19-5703858 

7-2621675 

•002610966 

384 

147456 

56623104 

19-5959179 

7-2684824 

•002604167 

385 

148225 

57066625 

19-6214169 

7-2747864 

•002597403 

386 

148996 

57512456 

19-6468827 

7-2810794 

•002590674 

387 

149769 

57960603 

19-6723156 

7-2873617 

•002583979 

388 

150544 

58411072 

19-6977156 

7-2936330 

•002577320 

389 

151321 

58863869 

19-7230829 

7-2998936 

•002570694 

390 

152100 

59319000 

19-7484177 

7-3061436 

•002564103 

391 

152881 

59776471 

19-7737199 

7-3123828 

•002557545 

392 

153664 

60236288 

19-7989899 

7-3186114 

•002551020 

393 

154449 

60698457 

19-8242276 

7-3248295 

•002544529 

394 

155236 

61162984 

19-8494332 

7-3310369 

•002538071 

395 

156025 

61629875 

19-8746069 

7-3372339 

•002531646 

396 

156816 

62099136 

19-8997487 

7-3434205 

•002525253 

397 

157609 

62570773 

19-9248588 

7-3495966 

•002518892 

398 

158404 

63044792 

19-9499373 

7-3557624 

•002512563 

399 

159201 

63521199 

19-9749844 

7-3619178 

•002506266 

400 

160000 

64000000 

20-0000000 

7-3680630 

•002500000 

401 

160801 

64481201 

20-0249844 

7-3741979 

•002493766 

402 

161604 

64964808 

20-0499377 

7-3803227 

•002487562 

403 

162409 

65450827 

20-0748599 

7-3864373 

•002481390 

404 

163216 

65939264 

20-0997512 

7-3925418 

•002475248 

405 

164025 

66430125 

20-1246118 

7-3986363 

•002469136 

406 

164836 

66923416 

20-1494417 

7-4047206 

•002463054 

407 

165649 

67419143 

20-1742410 

7-4107950 

•002457002 

408 

166464 

67917312 

20-1990099 

7-4168595 

•002450980 

409 

167281 

68417929 

20-2237484 

7-4229142 

•002444988 

410 

168100 

68921000 

20-2484567 

7-4289589 

•002439024 

411 

168921 

69426531 

20-2731349 

7-4349938 

•002433090 

412 

169744 

69934528 

20-2977831 

7-4410189 

•002427184 

413 

170569 

70444997 

20-3224014 

7-4470343 

•002421308 

414 

171396 

70957944 

20-3469899 

7-4530399 

•002415459 

415 

172225 

71473375 

20-3715488 

7-4590359 

•002409639 

416 

173056 

71991296 

20-3960781 

7-4650223 

•002406846 

417 

173889 

72511713 

20-4205779 

7-4709991 

•002398082 

418 

174724 

73034632 

20-4450483 

7-4769664 

•002392344 

419 

175561 

73560059 

20-4694895 

7-4829242 

•002386635 

420 

176400 

74088000 

20-4939015 

7-4888724 

•002380952 

421 

177241 

74618461 

20-5182845 

7-4948113 

•002375297 

422 

178084 

75151448 

20-5426386 

7-5007406 

•002369668 

423 

178929 

75686967 

20-5669638 

7-5066607 

•002364066 

424 

179776 

76225024 

20-5912603 

7-5125715 

•002358491 

425 

180625 

76765625 

20-6155281 

7-5184730 

•002352941 

426 

181476 

77308776 

20-6397674 

7-5243652 

•002347418 

427 

182329 

77854483 

20-6639783 

7-5302482 

•002341920 

428 

183184 

78402752 

20-6881609 

7-5361221 

•002336449 

429 

184041 

78953589 

20-7123152 

7-5419867 

•002331002 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.         107 


Number. 

Squares. 

Cubes.        Square  Roots. 

Cube  Roots. 

Reciprocals. 

430 

184900 

79507000 

20-7364414 

7-5478423 

•002325581 

431 

185761 

80062991 

20-7605395 

7-5536888 

•002320186 

432 

186624 

80621568 

20-7846097 

7-5595263  ' 

•002314815 

433 

187489 

81182737 

20-8086520 

7-5653548 

•002309469 

434 

188356 

81746504 

20-8326667 

7-5711743 

•002304147 

435 

189225 

82312875 

20-8566536 

7-5769849 

•002298851 

436 

190096 

82881856 

20-8806130 

7-5827865 

•002293578 

437 

190969 

83453453 

20-9045450 

7-5885793 

•002288330 

438 

191844 

84027672 

20-9284495 

7-5943633 

•002283105 

439 

192721 

84604519 

20-9523268 

7-6001385 

•002277904 

440 

193600 

85184000 

20-9761770 

7-6059049 

•002272727 

441 

194481 

85766121 

21-0000000 

7-6116626 

•002267574 

442 

195364 

86350888 

21-0237960 

7-6174116 

•002262443 

443 

196249 

86938307 

21-0475652 

7-6231519 

•002257336 

444 

197136 

87528384 

21-0713075 

7-6288837 

•002252252 

445 

198025 

88121125 

21-0950231 

7-6346067 

•002247191 

446 

"198916 

88716536 

21-1187121 

7-6403213 

•002242152 

447 

199809 

89314623 

21-1423745 

7-6460272 

•002237136 

448 

200704 

89915392 

21-1660105 

7-6517247 

•002232143 

449 

201601 

90518849 

21-1896201 

7-6574138 

•002227171 

450 

202500 

91125000 

21-2132034 

7-6630943 

•002222222 

451 

203401 

91733851 

21-2367606 

7-6687665 

•002217295 

452 

204304 

92345408 

21-2602916 

7-6744303 

•002212389 

453 

205209 

92959677 

21-2837967 

7-6800857 

•002207506 

454 

206116 

93576664 

21-3072758 

7-6857328 

•002202643 

455 

207025 

94196375 

21-3307290 

7-6913717 

•002197802 

456 

207936 

94818816 

21-3541565 

7-6970023 

•002192982 

457 

208849 

95443993 

21-3775583 

7-7026246 

•002188184 

458 

209764 

96071912 

21-4009346 

7-7082388 

•002183406 

459 

210681 

96702579 

21-4242853 

7-7188448 

•002178649 

460 

211600 

97336000 

21-4476106 

7-7194426 

•002173913 

461 

212521 

97972181 

21-4709106 

7-7250325 

•002169197 

462 

213444 

98611128 

21-4941853 

7-7306141 

•002164502 

463 

214369 

99252847 

21-5174348 

7-7361877 

•002159827 

464 

215296 

99897344 

21-5406592 

7-7417532 

•002155172 

465 

216225 

100544625 

21-5638587 

7-7473109 

•002150538 

466 

2L7156 

101194696 

21-5870331 

7-7528606 

•002145923 

467 

218089 

101847563 

21-6101828 

7-7584023 

•002141328 

468 

219024 

102503232 

21-6333077 

7-7639361 

•002136752 

469 

219961 

103161709 

21-6564078 

7-7694620 

•002132196 

470 

220900 

103823000 

21-6794834 

7-7749801 

•002127660 

471 

221841 

104487111 

21-7025344 

7-7804904 

•002123142 

472 

222784 

105154048 

21-7255610 

7-7859928 

•002118644 

473 

223729 

105828817 

21-7485632 

7-7914875 

•002114165 

474 

224676 

106496424 

21-7715411 

7-7969745 

•002109705 

475 

225625 

107171875 

21-7944947 

7-8024538 

•002105263 

476 

226576 

107850176 

21-8174242 

7-8079254 

•002100840 

477 

227529 

108531333 

21-8403297 

7-8133892 

•002096486 

478 

228484 

109215352 

21-8632111 

7-8188456 

•002092050 

479 

229441 

109902239 

21-8860686 

7-8242942 

•002087683 

480 

230400 

110592000 

21-9089023 

7-8297353 

•002083333 

481 

231361 

111284641 

21-9317122 

7-8351688 

•002079002 

482 

232324 

111980168 

21-9544984 

7-8405949 

•002074689 

483 

233289 

112678587 

21-9772610 

7-8460134 

•002070393 

484 

234256 

113379904 

22-0000000 

7-8514244 

•002066116 

485 

235225 

114084125 

22-0227155 

7-8568281 

•002061856 

486 

236196 

114791256 

22-0454077 

7-8622242 

•002057613 

487 

237169 

115501303 

22-0680765 

7-8676130 

•002053388 

488 

238144 

116214272 

22-0907220 

7-8729944 

•002049180 

489 

239121 

116930169 

22-1133444 

7-8783684 

•002044990 

490 

240100 

117649000 

22-1359436 

7-8837352 

•002040816 

491 

241081 

118370771 

22-1585198 

7-8890946 

•002036660 

108 


THE    PRACTICAL   MODEL   CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

492 

242064 

119095488 

22-1810730 

7-8944468 

•002032520 

493 

243049 

119823157 

22-2036033 

7-8997917 

•002028398 

494 

244036 

120553784 

22-2261108 

7-9051294 

•002024291 

495 

245025 

121287375 

22-2485955 

7-9104599 

•002020202 

496 

246016 

122023936 

22-2710575 

7-9157832 

•002016129 

497 

247009 

122763473 

22-2934968 

7-9210994 

•002012072 

498 

248004 

123505992 

22-3159136 

7-9264085 

•002008032 

499 

249001 

124251499 

22-3383079 

7-9317104 

•002004008 

500 

250000 

125000000 

22-3606798 

7-9370053 

•002000000 

501 

251001 

125751501 

22-3830293 

7-9422931 

•001996008 

502 

252004 

126506008 

22-4053565 

7-9475739 

•001992032 

503 

253009 

127263527 

22-4276615 

7-9528477 

•001988072 

504 

254016 

128024064 

22-4499443 

7-9581144 

•001984127 

505 

255025 

128787625 

22-4722051 

7-9633743 

•001980198 

506 

256036 

129554216 

22-4944438 

7-9686271 

•001976285 

507 

257049 

130323843 

22-5166605 

7-9738731 

•001972387 

508 

258064 

131096512 

22-5388553 

7-9791122 

•0*01968504 

509 

259081 

131872229 

22-5610283 

7-9843444 

•001964637 

510 

260100 

132651000 

22-5831796 

7-9895697 

•001960784 

511 

261121 

133432831 

22-6053091 

7-9947883 

•001956947 

512 

262144 

134217728 

22-6274170 

8-0000000 

•001953125 

513 

263169 

135005697 

22-6495033 

8-0052049 

•001949318 

514 

264196 

135796744 

22-6715681 

8-0104032 

•001945525 

515 

265225 

136590875 

22-6936114 

8-0155946 

•001941748 

516 

266256 

137388096 

22-7156334 

8-0207794 

•001937984 

517 

267289 

138188413 

22-7376341 

8-0259574 

•001934236 

518 

268324 

138991832 

22-7596134 

8-0311287 

•001930502 

519 

269361 

139798359 

22-7815715 

8-0362935 

•001926782 

520 

270400 

140608000 

22-8035085 

8-0414515 

•001923077 

521 

271411 

141420761 

22-8254244 

8-0466030 

•001919386 

522 

272484 

142236648 

22-8473193 

8-0517479 

•001915709 

523 

273529 

143055667 

22-8691933 

8-0568862 

•001912046 

524 

274576 

143877824 

22-8910463 

8-0620180 

•001908397 

525 

275625 

144703125 

22-9128785 

8-0671432 

•001904762 

526 

276676 

145531576 

22-9346899 

8-0722620 

•001901141 

527 

277729 

146363183 

22-9564806 

8-0773743 

•001897533 

528 

278784 

147197952 

22-9782506 

8-0824800 

•001893939 

529 

279841 

148035889 

23-0000000 

8-0875794 

•001890359 

530 

280900 

148877001 

23-0217289 

8-0926723 

•001886792 

531 

281961 

149721291 

23-0434372 

8-0977589 

•001883239 

532 

283024 

150568768 

23-0651252 

8-1028390 

•001879699 

533 

284089 

151419437 

23-0867928 

8-1079128 

•001876173 

534 

285156 

152273304 

23-1084400 

8-1129803 

•001872659 

535 

286225 

153130375 

23-1300670 

8-1180414 

•001869159 

536 

287296 

153990656 

23-1516738 

8-1230962 

•001865672 

537 

288369 

154854153 

23-1732605 

8-1281447 

•001862197 

538 

289444 

155720872 

23-1948270 

8-1331870 

•001858736 

539 

290521 

156590819 

23-2163735 

8-1382230 

•001855288 

540 

291600 

157464000 

23-2379001 

8-1432529 

•001851852 

641 

292681 

158340421 

23-2594067 

8-1482765 

•001848429 

542 

293764 

159220088 

23-2808935 

8-1532939 

•001845018 

543 

294849 

160103007 

23-3023604 

8-1583051 

•001841621 

544 

295936 

160989184 

23-3238076 

8-1633102 

•001838235 

545 

297025 

161878625 

23-3452351 

8-1683092 

•001834862 

546 

298116 

162771336 

23-3666429 

8-1733020 

•001831502 

647 

299209 

163667323 

23-3880311 

8-1782888 

•001828154 

548 

300304 

164566592 

23-4093998 

8-1832695 

•001824818 

549 

301401 

165469149 

23-4307490 

8-1882441 

•001821494 

550 

302500 

166375000 

23-4520788 

8-1932127 

•001818182 

551 

303601 

167284151 

23-4733892 

8-1981753 

•001814882 

552 

304704 

168196608 

23-4946802 

8-2031319 

•001811594 

553 

305809 

169112377 

23-5159520 

8-2080825 

•001808318 

TABLE  OP  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.         109 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

554 

306916 

170031464 

23-5372046 

8-2130271 

•001805054 

555 

308025 

170953875 

23-5584380 

8-2179657 

•001801802 

556 

309136 

171879616 

23-5796522 

8-2228985 

•001798561 

557 

310249 

172808693 

23-6008474 

8-2278254 

•001795332 

558 

311364 

173741112 

23-6220236 

8-2327463 

•001792115 

559 

312481 

174676879 

23-6431808 

8-2376614 

•001788909 

560 

313600 

175616000 

23-6643191 

8-2425706 

•001785714 

561 

314721 

176558481 

23-6854386 

8-2474740 

•001782531 

562 

315844 

177504328 

23-7065392 

8-2523715 

•001779359 

563 

316969 

178453547 

23-7276210 

8-2572635 

•001776199 

564 

318096 

179406144 

23-7486842 

8-2621492 

•001773050 

565 

319225 

180362125 

23-7697286 

8-2670294 

•001769912 

566 

320356 

181321496 

23-7907545 

8-2719039 

•001766784 

567 

321489 

182284263 

23-8117618 

8-2767726 

•001763668 

568 

322624 

183250432 

23-8327506 

8-2816255 

•001760563 

569 

323761 

184220009 

23-8537209 

8-2864928 

•001757469 

570 

324900 

185193000 

23-8746728 

8-2913444 

•001754386 

571 

326041 

186169411 

23-8956063 

8-2961903 

•001751313 

572 

327184 

187149248 

23-9165215 

8-3010304 

•001748252 

573 

328329 

188132517 

23-9374184 

8-3058651 

•001745201 

574 

329476 

189119224 

23-9582971 

8-3106941 

•001742160 

575 

330625 

190109375 

23-9791576 

8-3155175 

•001739130 

576 

331776 

191102976 

24-0000000 

8-3203353 

•001736111 

577 

332927 

192100033 

24-0208243 

8-3251475 

•001733102 

578 

334084 

193100552 

24-0416306 

8-3299542 

•001730104 

579 

335241 

194104539 

24-0624188 

8-3347553 

•001727116 

580 

336400 

195112000 

24-0831891 

8-3395509 

•001724138 

581 

337561 

196122941 

24-1039416 

8-3443410 

•001721170 

582 

338724 

197137368 

24-1246762 

8-3491256 

•001718213 

583 

339889 

198155287 

24-1453929 

8-3539047 

•001715266 

584 

341056 

199176704 

24-1660919 

8-3586784 

•001712329 

585 

342225 

200201625 

24-1867732 

8-3634466 

•001709402 

586 

343396 

201230056 

24-2074369 

8-3682095 

•001706485 

587 

344569 

202262003 

24-2280829 

8-3729668 

•001703578 

588 

345744 

203297472 

24-2487113 

8-3777188 

•001700680 

589 

346921 

204336469 

24-2693222 

8-3824653 

•001697793 

590 

348100 

205379000 

24-2899156 

8-3872065 

•001694915 

591 

349281 

206425071 

24-3104996 

8-3919428 

•001692047 

592 

350464 

207474688 

24-3310501 

8-3966729 

•001689189 

593 

351649 

208527857 

24-3515913 

8-4013981 

•001686341 

594 

352836 

209584584 

24-3721152 

8-4061180 

•001683502 

595 

354025 

210644875 

24-3926218 

8-4108326 

•001680672 

596 

355216 

211708736 

24-4131112 

8-4155419 

•001677852 

597 

356409 

212776173 

24-4335834 

8-4202460 

•001675042 

598 

357604 

213847192 

24-4540385 

8-4249448 

•001672241 

599 

358801 

214921799 

24-4744765 

8-4296383 

•001669449 

600 

360000 

216000000 

24-4948974 

8-4343267 

•001666667 

601 

361201 

217081801 

24-5153013 

8-4390098 

•001663894 

602 

362404 

218167208 

24-5356883 

8-4436877 

•001661130 

603 

363609 

219256227 

24-5560583 

8-4483605 

•001658375 

604 

364816 

220348864 

24-5764115 

8-4530281 

•001655629 

605 

366025 

221445125 

24-5967478 

8-4576906 

•001652893 

606 

367236 

222545016 

24-6170673 

8-4623479 

•001650165 

607 

368449 

223648543 

24-6373700 

8-4670001 

•001647446 

608 

369664 

224755712 

24-6576560 

8-4716471 

•001644737 

609 

370881 

225866529 

24-6779254 

8-4762892 

•001642036 

610 

372100 

226981000 

24-6981781 

8-4809261 

•001639344 

611 

373321 

228099131 

24-7184142 

8-4855579 

•001636661 

612 

374544 

229220928 

24-7386338 

8-4901848 

•001633987 

613 

375769 

230346397 

24-7588368 

8-4948065 

•001631321 

614 

376996 

231475544 

24-7790234 

8-4994233 

•001628664 

615 

378225 

232608375 

24-7991935 

8-5040350 

•001626016 

110 


THE   PRACTICAL   MODEL    CALCULATOR. 


Number. 

Squares.        Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

616 

379456 

233744896 

24-8193473 

8-5086417 

•001623377 

617 

380689 

2348851,13 

24-8394847 

8-5132435 

•001620746 

618 

381924 

236029032 

24-8596058 

8-5178403 

•001618123 

619 

383161 

237176659 

24-8797106 

8-5224331 

•001615509 

620 

384400 

238328000 

24-8997992 

8-5270189 

•001612903 

621 

385641 

239483061 

24-9198716 

8-5316009 

•001610306 

622 

386884 

240641848 

24-9399278 

8-5361780 

•001607717 

623 

388129 

241804367 

24-9599679 

8-5407501 

•001605136 

624 

389376 

242970624 

24-9799920 

8-5453173 

•001602564 

625 

390625 

244140625 

25-0000000 

8-5498797 

•001600000 

626 

391876 

245134376 

25-0199920 

8-5544372 

•001597444 

627 

393129 

246491883 

25-0399681 

8-5589899 

•001594896 

628 

.  394384 

247673152 

25-0599282 

8-5635377 

•001592357 

629 

395641 

248858189 

25-0798724 

8-5680807 

•001589825 

630 

396900 

250047000 

25-0998008 

8-5726189 

•001  587302 

631 

398161 

251239591 

25-1197134 

8-5771523 

•001584786 

632 

399424 

252435968 

25-1396102 

8-5816809 

•001582278 

633 

400689 

253636137 

25-1594913 

8-5862247 

•001579779 

634 

401956 

254840104 

25-1793566 

8-5907238 

•001577287 

635 

403225 

256047875 

25-1992063 

8-5952380 

•001574803 

636 

404496 

257259456 

25-2190404 

8-5997476 

•001572327 

637 

405769 

258474853 

25-23-88589 

8-6042525 

•0015G9859 

638 

407044 

259694072 

25-2586619 

8-6087526 

•001567398 

639 

408321 

260917119 

25-2784493 

8-6132480 

•001564945 

640 

409600 

262144000 

25-2982213 

8-6177388 

•001562500 

641 

410881 

263374721 

25-3179778 

8-6222248 

•001560062 

642 

412164 

264609288 

25-3377189 

8-6267063 

•001557632 

643 

413449 

265847707 

23-3574447 

8-6311830 

•001555210 

644 

414736 

267089984- 

25-3771551 

8-6356551 

•001552795 

645 

416125 

268336125 

25-3968502 

8-6401226 

•001550388 

646 

417316 

269585136 

254165302 

8-6445855 

•001547988 

647 

418609 

270840023 

25-4361947 

8-6490437 

•001545595 

648 

419904 

272097792 

25-4558441 

8-6534974 

•001543210 

649 

421201 

273359449 

254754784 

8-6579465 

•001540832 

650 

422500 

274625000 

254950976 

8-6623911 

•001538462 

651 

423801 

275894451 

25-5147013 

8-6668310 

•001536098 

652 

425104 

277167808 

25-5342907 

8-6712665 

•001533742 

653 

426409 

278445077 

25-5538647 

8-6756974 

•001531394 

654 

427716 

279726264 

25-5734237 

8-6801237 

•001529052 

655 

429025 

281011375 

25-5929678 

8-6845456 

•001526718 

656 

430336 

282300416 

25-6124969 

8-6889630 

•001524390 

657 

431639 

283593393 

25-6320112 

8-6933759 

•001522070 

658 

432964 

284890312 

25-6515107 

8-6977843 

•001519751 

659 

434281 

286191179 

25-6709953 

8-7021882 

•001517451 

660 

435600 

287496000 

25-6904652 

8-7065877 

•001515152 

661 

436921 

288804781 

25-7099203 

8-7109827 

•001512859 

662 

438244 

290117528 

25-7293607 

8-7153734 

•001510574 

663 

489569 

291434247 

25-7487864 

8-7197596 

•001508296 

664 

440896 

292754944 

25-7681975 

8-7241414 

•001506024 

665 

442225 

294079625 

25-7875939 

8-7285187 

•001503759 

666 

443556 

295408296 

25-8069758 

8-7328918 

•001501502 

667 

444899 

296740963 

25-8263431 

8-7372604 

•001499250 

668 

446224 

298077632 

25-8456960 

8-7416246 

•001497006 

669 

447561 

299418309 

25-8650343 

8-7459846 

•001494768 

670 

448900 

300763000 

25-8843582 

8-7503401 

•001492537 

671 

450241 

302111711 

25-9036677 

8-7546913 

•001490313 

672 

451584 

303464448 

25-9229628 

8-7590383 

•001488095 

673 

462929 

304821217 

25-9422435 

8-7633809 

•001485884 

674 

454276 

306182024 

25-9615100 

8-7677192 

•001483680 

675 

455625 

307546875 

25-9807621 

8-7720532 

•001481481 

676 

466976 

308915776 

26-0000000 

8-7763830 

•001479290 

677 

458329 

310288733 

26-0192237 

8-7807084 

•001477105 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.          Ill 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

678 

459684 

311665752 

26-0384331 

8-7850296 

•001474926 

679 

461041 

313046839 

26-0576284 

8-7893466 

•001472754 

680 

462400 

314432000 

26-0768096 

8-7936593 

•001470588 

681 

463761 

315821241 

26-0959767 

8-7979679 

•001468429 

682 

465124 

317214568 

26-1151297 

8-8022721 

•001466276 

683 

466489 

318611987 

26-1342687 

8-8065722 

•001464129 

684 

467856 

320013504 

26-1533937 

8-8108681 

•001461988 

685 

469225 

321419125 

26-1725047 

8-8151598 

•001459854 

686 

470596 

322828856 

26-1916017 

8-8194474 

•001457726 

687 

471969 

324242703 

26-2106848 

8-8237307 

•001455604 

688 

473344 

325660672 

26-2297541 

8-8280099 

•001453488 

689 

474721 

327082769 

26-2488095 

8-8322850 

•001451379 

690 

476100 

328509000 

26-2678511 

8-8365559 

•001449275 

691 

477481 

329939371 

26-2868789 

8-8408227 

•001447178 

692 

478864 

331373888 

26-3058929 

8-8450854 

•001445087 

693 

480249 

332812557 

26-3248932 

8-8493440 

•001443001 

694 

481636 

334255384 

26-3438797 

8-8535985 

•001440922 

695 

483025 

335702375 

26-3628527 

8-8578489 

•001438849 

696 

484416 

337153536 

26-3818119 

8-8620952 

•001436782 

697 

485809 

338608873 

26-4007576 

8-8663375 

•001434720 

698 

487204 

340068392 

26-4196896 

8-8705757 

•001432665 

699 

488601 

341532099 

26-4386081 

'8-8748099 

•001430615 

700 

490000 

343000000 

26-4575131 

8-8790400 

•001428571 

701 

491401 

344472101 

26-4764046 

8-8832661 

•001426534 

702 

492804 

345948408 

26-4952826 

8-8874882 

•001424501 

703 

494209 

347428927 

26-5141472 

8-8917063 

•001422475 

704 

495616 

348913664 

26-5329983 

8-8959204 

•001420455 

705 

497025 

350402625 

26-5518361 

8-9001304 

•001418440 

706 

498436 

351895816 

26-5706605 

8-9043366 

•001416431 

707 

499849 

353393243 

26-5894716 

8-9085387 

•001414427 

708 

501264 

354894912 

26-6082694 

8-9127369 

•001412429 

709 

502681 

356400829 

26-6270539 

8-9169311 

•001410437 

710 

504100 

357911000 

26-6458252 

8-9211214 

•001408451 

711 

505521 

359425431 

26-6645833 

8-9253078 

•001406470 

712 

506944 

360944128 

26-6833281 

8-9294902 

•001404494 

713 

508369 

362467097 

26-7020598 

8-9336687 

•001402525 

714 

509796 

363994344 

26-7207784 

8-9378433 

•001400560 

715 

511225 

365525875 

26-7394839 

8-9420140 

•001398601 

716 

512656 

367061696 

26-7581763 

8-9461809 

•001396648 

717 

514089 

368601813 

26-7768557 

8-9503438 

•001394700 

718 

515524 

370146232 

26-7955220 

8-9545029 

•001392758 

719 

516961 

371694959 

26-8141754 

8-9586581 

•001390821 

720 

518400 

373248000 

26-8328157 

8-9628095 

•001388889 

721 

519841 

374805361 

26-8514432 

8-9669570 

•001386963 

722 

521284 

376367048 

26-8700577 

8-9711007 

•001385042 

723 

522729 

377933067 

26-8886593 

8-9752406 

•001383126 

724 

524176 

379503424 

26-9072481 

8-9793766 

•001381215 

725 

525625 

381078125 

26-9258240 

8-9835089 

•001379310 

726 

527076 

382657176 

26-9443872 

8-9876373 

•001377410 

727 

528529 

384240583 

26-9629375 

8-9917620 

•001375516 

728 

529984 

385828352 

26-9814751 

8-9958899 

•001373626 

729 

531441 

387420489 

27-0000000 

9-0000000 

•001371742 

730 

532900 

389017000 

27-0185122 

9-0041134 

•001369863 

731 

534361 

390617891 

27-0370117 

9-0082229 

•001367989 

732 

535824 

392223168 

27-0554985 

9-0123288 

•001366120 

733 

537289 

393832837 

27-0739727 

9-0164309 

•001364256 

734 

538756 

395446904 

27-0924344 

9-0205293 

•001362398 

735 

540225 

397065375 

27-1108834 

9-0246239 

•001360544 

736 

541696 

398688256 

27-1293199 

9-0287149 

•001358G96 

737 

543169 

400315553 

27-1477149 

9-0328021 

•001356852 

738 

544644 

401947272 

27-1661554 

9-0368857 

•001355014 

739 

546121 

403583419 

27-1845544 

9-0409655 

•001353180 

112 


THE  PRACTICAL  MODEL  CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

740 

547600 

405224000 

27-2029140 

9-0450419 

•001351351 

741 

549801 

406869021 

27-2213152 

9-0491142 

•001349528 

742 

550564  ' 

408518488 

27-2396769 

9-0531831 

•001347709 

743 

552049 

410172407 

27-2580263 

9-0572482 

•001345895 

744 

553536 

411830784 

27-2763634 

9-0613098 

•001344086 

745 

555025 

413493625 

27-2946881 

9-0653677 

•001342282 

746 

656516 

415160936 

27-3130006 

9-0694220 

•001340483 

747 

558009 

416832723 

27-3313007 

9-0734726 

•001338688 

748 

659504 

418508992 

27-3495887 

9-0775197 

•001336898 

749 

561001 

420189749 

27-3678644 

9-0815631 

•001335113 

750 

562500 

421875000 

27-3861279 

9-0856030 

•001333333 

751 

564001 

423564751 

27-4043792 

9-0896352 

•001331558 

752 

565504 

425259008 

27-4226184 

9-0936719 

•001329787 

753 

567009 

426957777 

27-4408455 

9-0977010 

•001328021 

754 

668516 

428661064 

27-4590604 

9-1017265 

•001326260 

755 

570025 

430368875 

27-4772633 

9-1057485 

•001324503 

756 

571536 

432081216 

27-4954542 

9-1097669 

•001322751 

757 

673049 

433798093 

27-5136330 

9-1137818 

•001321004 

758 

574564 

435519512 

27-5317998 

9-1177931 

•001319261 

759 

576081 

437245479 

27-5499546 

9-1218010 

•001317523 

760 

577600 

438976000 

27-5680975 

9-1258053 

•001315789 

761 

579121 

440711081 

27-5862284 

9-1298061 

•001314060 

762 

580644 

442450728 

27-6043475 

9-1338034 

•001312336 

763 

582169 

444194947 

27-6224546 

9-1377971 

•001310616 

764 

583696 

445943744 

27-6405499 

9-1417874 

•001308901 

765 

585225 

447697125 

27-6586334 

9-1457742 

•001307190 

766 

586756 

449455096 

27-6767050 

9-1497576 

•001305483 

767 

588289 

451217663 

27-6947648 

9-1537375 

•001303781 

768 

589824 

452984832 

27-7128129 

9-1577139 

•001302083 

769 

591361 

454756609 

27-7308492 

9-1616869 

•001300390 

770 

592900 

456533000 

27-7488739 

9-1656565 

•001298701 

771 

594441 

458314011 

27-7668868 

9-1696225 

•001297017 

772 

595984 

460099648 

27-7848880 

9-1735852 

•001295337 

773 

597529 

461889917 

27-8028775 

9-1775445 

•001293661 

774 

699076 

463684824 

27-8208555 

9-1815003 

•001291990 

775 

600625 

465484375 

27-8388218 

9-1854527 

•001290323 

776 

602176 

467288576 

27-8567766 

9-1894018 

•001288660 

777 

603729 

469097433 

27-8747197 

9-1933474 

•001287001 

778 

605284 

470910952 

27-8926514 

9-1972897 

•001285347 

779 

606841 

472729139 

27-9105715 

9-2012286 

•001283697 

780 

608400 

474552000 

27-9284801 

9-2051641 

•001282051 

781 

609961 

476379541 

27-9463772 

9-2090962 

•001280410 

782 

611524 

478211768 

27-9642629 

9-2130250 

•001278772 

783 

613089 

480048687 

27-9821372 

9-2169505 

•001277139 

784 

614656 

481890304 

28-0000000 

9-2208726 

•001275510 

785 

616225 

483736625 

28-0178515 

9-2247914 

•001273885 

786 

617796 

485587656 

28-0356915 

9-2287068 

•001272265 

787 

619369 

487443403 

28-0535203 

9-2326189 

•001270648 

788 

620944 

489303872 

28-0713377 

9-2365277 

•001269036 

789 

622521 

491169069 

28-0891438 

9-2404333 

•0012G7427 

790 

624100 

493039000 

28-1069386 

9-2443355 

•001265823 

791 

625681 

494913671 

28-1247222 

9-2482344 

•001264223 

792 

627624 

496793088 

28-1424946 

8-2521300 

•001262626 

793 

628849 

498677257 

28-1602557 

9-2560224 

•001261034 

794 

630436 

500566184 

28-1780056 

9-2599114 

•001259446 

795 

632025 

502459875 

28-1957444 

9-2637973 

•001257862 

796 

633616 

504358336 

28-2134720 

9-2676798 

•001256281 

797 

635209 

506261573 

28-2311884 

9-2715592 

•001254705 

798 

636804 

508169592 

28-2488938 

9-2754352 

•001253133 

799 

638401 

510082399 

28-2665881 

9-2793081 

•001251364 

800 

640000 

512000000 

28-2842712 

9-2831777 

•001250000 

801 

641601 

513922401 

28-3019434 

9-2870444 

•001248439 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    113 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

802 

643204 

515849608 

28-3196045 

9-2909072 

•001246888 

803 

644809 

517781627 

28-3372546 

9-2947671 

•001245330 

804 

646416 

519718464 

28-3548938 

9-2986239 

•001243781 

805 

648025 

521660125 

28-3725219 

9-3024775 

•001242236 

806 

649636 

523606616 

28-3901391 

9-3063278 

•001240695 

807 

651249 

525557943 

28-4077454 

9-3101750 

•001239157 

808 

652864 

527514112 

28-4253408 

9-3140190 

•001237624 

809 

654481 

529475129 

28-4429253 

9-3178599 

•001236094 

810 

656100 

531441000 

28-4604989 

9-3216975 

•001234568 

811 

657721 

533411731 

28-4780617 

9-3255320 

•001233046 

812 

659344 

535387328 

28-4956137 

9-3293634 

•001231527 

813 

660969 

537367797 

28-5131549 

9-3331916 

•001230012 

814 

662596 

539353144 

28-5306852 

9-3370167 

•001228501 

815 

664225 

541343375 

28-5482048 

9-3408386 

•001226994 

816 

665856 

543338496 

28-5657137 

9-3446575 

•001225499 

817 

667489 

545338513 

28-5832119 

9-3484731 

'•001223990 

818 

669124 

547343432 

28-6006993 

9-3522857 

•001222494 

819 

670761 

549353259 

28-6181760 

9-3560952 

•001221001 

820 

672400 

551368000 

28-6356421 

9-3599016 

•001219512 

821 

674041 

553387661 

28-6530976 

9-3637049 

•001218027 

822 

675684 

555412248 

28-6705424 

9-3675051 

•001216545 

823 

677329 

557441767 

28-6879716 

9-3713022 

•001215067 

824 

678976 

559476224 

28-7054002 

9-3750963 

•001213592 

825 

680625 

561515625 

28-7228132 

9-3788873 

•001212121 

826 

682276 

563559976 

28-7402157 

9-3826752 

•001210654 

827 

683929 

565609283 

28-7576077 

9-3864600 

•001209190 

828 

685584 

567663552 

28-7749891 

9-3902419 

•001207729 

829 

687241 

569722789 

28-7923601 

9-3940206 

•001206273 

830 

688900 

571787000 

28-8097206 

9-3977964 

•001204819 

831 

690561 

573856191 

28-8270706 

9-4015691 

•001203369 

832 

692224 

575930368 

28-8444102 

9-4053387 

•001201923 

833 

693889 

578009537 

28-8617394 

9-4091054 

•001200480 

834 

695556 

580093704 

28-8790582 

9-4128690 

•001199041 

835 

697225 

582182875 

28-8963666 

9-4166297 

•001197605 

836 

698896 

584277056 

28-9136646 

9-4203873 

•001196172 

837 

700569 

586376253 

28-9309523 

9-4241420 

•001194743 

838 

702244 

588480472 

28-9482297 

9-4278936 

•001193317 

839 

703921 

590589719 

28-9654967 

9-4316423 

•001191895 

840 

705600 

592704000 

28-9827535 

9-4353800 

•001190476 

841 

707281 

594823321 

29-0000000 

9-4391307 

•001189061 

842 

708964 

596947688 

29-0172363 

9-4428704 

•001187648 

843 

'  710649 

599077107 

29-0344623 

9-4466072 

•001186240 

844 

712336 

601211584 

29-0516781 

9-4503410 

•001184834 

845 

714025 

603351125 

29-0688837 

9-4540719 

•001183432 

846 

715716 

605495736 

29-0860791 

9-4577999 

•001182033 

847 

717409 

607645423 

29-1032644 

9-4615249 

•001180638 

848 

719104 

609800192 

29-1204396 

9-4652470 

•001179245 

849 

720801 

611960049 

29-1376046 

9-4689661 

•001177856 

850 

722500 

614125000 

29-1547595 

9-4726824 

•001176471 

851 

724201 

616295051 

29-1719043 

9-4763957 

•001175088 

852 

725904 

618470208 

29-1890390 

9-4801061 

•001173709 

853 

727609 

620650477 

29-2061637 

9-4838136 

•001172333 

854 

729316 

622835864 

29-2232784 

9-4875182 

•001170960 

855 

731025 

625026375 

29-2403830 

9-4912200 

•001169591 

856 

732736 

627222016 

29-2574777 

9-4949188 

•001168224 

857 

734449 

629422793 

29-2745623 

9-4986147 

•001166861 

858 

736164 

631628712 

29-2916370 

9-5023078 

•001165501 

859 

737881 

633839779 

29-3087018 

9-5059980 

•001164144 

860 

739600 

636056000 

29-3257566 

9-5096854 

•001162791 

861 

741321 

638277381 

29-3428015 

9-5133699 

•001161440 

862 

743044 

640503928 

29-3598365 

9-5170515 

•001160093 

863 

744769 

642735647 

29-3768616 

9-5207303 

•001158749 

114 


THE   PRACTICAL    MODEL   CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

864 

746496 

644972544 

29-3938769 

9-5244063 

•001157407 

865 

748225 

647214625 

29-4108823 

9-5280794 

•001156069 

866 

749956 

649461896 

29-4278779 

9-5317497 

•001154734 

867 

751689 

651714363 

29-4448637 

9-5354172 

•001153403 

868 

753424 

653972032 

29-4618397 

9-5390818 

•001152074 

869 

755161 

656234909 

29-4788059 

9-5427437 

•001150748 

870 

756900 

658503000 

29-4957624 

9-5464027 

•001149425 

871 

758641 

660776311 

29-5127091 

9-5500589 

•001148106 

872 

760384 

663054848 

29-5296461 

9-5537123 

•001146789 

873 

762129 

665338617 

29-5465734 

9-5573630 

•001145475 

874 

763876 

667627624 

29-5634910 

9-5610108 

•001144165 

875 

765625 

669921875 

29-5803989 

9-5646559 

•001142857 

876 

767376 

672221376 

29-5972972 

9-5682782 

•001141553 

877 

769129 

674526133 

29-6141858 

9-5719377 

•001140251 

878 

7.70884 

676836152 

29-6310648 

9-5755745 

•001138952 

879 

772641 

679151439 

29-6479342 

9-5792085 

•001137656 

880 

774400 

681472000 

29-6647939 

9-5828397 

•001136364 

881 

776161 

683797841 

29-6816442 

9-5864682 

•001135074 

882 

777924 

686128968 

29-6984848 

9-5900937 

•001133787 

883 

779689 

688465387 

29-7153159 

9-5937169 

•001132503 

884 

781456 

690807104 

29-7321375 

9-5973373 

•001131222 

885 

783225 

693154125 

29-7489496 

9-6009548 

•001129944 

886 

784996 

695506456 

29-7657521 

9-6045696 

•001128668 

887 

786769 

697864103 

29-7825452 

9-6081817 

•001127396 

888 

788544 

700227072 

29-7993289 

9-6117911 

•001126126 

889 

790321 

702595369 

29-8161030 

9-6153977 

•001124859 

890 

792100 

704969000 

29-8328678 

9-6190017 

•001123596 

891 

793881 

707347971 

29-8496231 

9-6226030 

•001122334 

892 

795664 

707932288 

29-8663690 

9-6262016 

•001121076 

893 

797449 

712121957 

29-8831056 

9-6297975 

•001119821 

894 

799236 

714516984 

29-8998328 

9-6333907 

•001118568 

895 

801025 

716917375 

29-9165506 

9-6369812 

•001117818 

896 

802816 

719323136 

29-9332591 

9-6405690 

•001116071 

897 

804609 

721734273 

29-9499583 

9-6441542 

•001114827 

898 

806404 

724150792 

29-9666481 

9-6477367 

•001113586 

899 

808201 

726572699 

29-9833287 

9-6513166 

•001112347 

900 

810000 

729000000 

30-0000000 

9-6548938 

•001111111 

901 

811801 

731432701 

30-0166621 

9-6584684 

•001109878 

902 

813604 

733870808 

30-0333148 

9-6620403 

•001108647 

903 

815409 

736314327 

30-0499584 

9-6656096 

•001107420 

504 

817216 

738763264 

30-0665928 

9-6691762 

•001106195 

905 

819025 

741217625 

30-0832179 

9-6727403 

•001104972 

906 

820836 

743677416 

30-0998339 

9-6763017 

•001103753 

907 

822649 

746142643 

30-1164407 

9-6798604 

•001102536 

908 

824464 

748613312 

30-1330383 

9-6834166 

•001101322 

909 

826281 

751089429 

30-1496269 

9-6869701 

•001100110 

910 

828100 

753571000 

30-1662063 

9-6905211 

•001098901 

911 

829921 

756058031 

30-1827765 

9-6940694 

•001097695 

912 

831744 

758550825 

30-1993377 

9-6976151 

•001096491 

913 

833569 

761048497 

30-2158899 

9-7011583 

•001095290 

914 

835396 

763551944 

30-2324329 

9-7046989 

•001094092 

915 

837225 

766060875 

30-2489669 

9-7082369 

•001092896 

916 

839056 

768575296 

30-2654919 

9-7117723 

•001091703 

917 

840889 

771095213 

30-2820079 

9-7153051 

•001090513 

918 

842724 

773620632 

30-2985148 

9-7188354 

•001089325 

919 

844561 

776151559 

30-3150128 

9-7223631 

•001088139 

920 

846400 

778688000 

30-3315018 

9-7258883 

•001086957 

921 

848241 

781229961 

30-3479818 

9-7294109 

•001085776 

922 

850084 

783777448 

30-3644529 

9-7329309 

•001084599 

923 

851929 

786330467 

30-3809151 

9-7364484 

•001083423 

924 

853776 

788889024 

30-3973683 

9-7399634 

•001082251 

925 

855625 

791453125 

30-4138127 

9-7434758 

•001081081 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    115 


Niii  ber.    Squares. 

Cubes, 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

926 

857476 

794022776 

30-4302481 

9-7469857 

•001079914 

927 

859329 

796597983 

30-4466747 

9-7504930 

•001078749 

928 

861184 

799178752 

30-4630924 

9-7539979 

•001077686 

929 

863041 

801765089 

30-4795013 

9-7575002 

•001076426 

930 

864900 

804357000 

30-4959014 

9-7610001 

•001075269 

931 

866761 

806954491 

30-5122926 

9-7644974 

•001074114 

932 

868624 

809557568 

30-5286750 

9-7679922 

•001072961 

933 

870489 

812166237 

30-5450487 

9-7714845 

•001071811 

934 

872356 

814780504 

30-5614136 

9-7749743 

•001070664 

935 

874225 

817400375 

30-5777697 

9-7784616 

•001069519 

936 

876096 

820025856 

30-5941171 

9-7829466 

•001068376 

937 

877969 

822656953 

30-6104557 

9-7854288 

•001067236 

938 

879844 

825293672 

30-6267857 

9-7889087 

•001066098 

939 

881721 

827936019 

30-6431069 

9-7923861 

•001064963 

940 

883600 

830584000 

30-6594194 

9-7958611 

•001063830 

941 

885481 

833237621 

30-6757233 

9-7993336 

•001062699 

942 

887364 

835896888 

30-6920185 

9-8028036 

•001061571 

943 

889249 

838561807 

30-7083051 

9-8062711 

•001060445 

944 

891136 

841232384 

30-7245830 

9-8097362 

•001059322 

945 

893025 

843908625 

30-7408523 

9-8131989 

•001058201 

946 

894916 

846590536 

30-7571130 

9-8166591 

•001057082 

947 

896808 

849278123 

30-7733651 

9-8201169 

•001055966 

948 

898704 

851971392 

30-7896086 

9-8235723 

•001054852 

949 

900601 

854670349 

30-8058436 

9-8270252 

•001053741 

950 

902500 

857375000 

30-8220700 

9-8304757 

•001052632 

951 

904401 

860085351 

30-8382879 

9-8339238 

•001051525 

952 

906304 

862801408 

30-8544972 

•  9-8373695 

•001050420 

953 

908209 

865523177 

30-8706981 

9-8408127 

•001049318 

954 

910116 

868250664 

30-8868904 

9-8442536 

•001048218 

955 

912025 

870983875 

30-9030743 

9-8476920 

•001047120 

956 

913936 

873722816 

30-9192477 

9-8511280 

•001046025 

957 

915849 

876467493 

30-9354166 

9-8545617 

•001044932 

958 

917764 

879217912 

30-9515751 

9-8579929 

•001043841 

959 

919681 

881974079 

30-9677251 

9-8614218 

•001042753 

960 

921600 

884736000 

30-9838668 

9-8648483 

•001041667 

961 

923521 

887503681 

31-0000000 

9-8682724 

•001040583 

962 

925444 

890277128 

31-0161248 

9-8716941 

•001039501 

963 

927369 

893056347 

31-0322413 

9-8751135 

•001038422 

964 

929296 

895841344 

31-0483494 

9-8785305 

•001037344 

965 

931225 

898632125 

31-0644491 

9-8819451 

•001036269 

966 

933156 

901428696 

31-0805405 

9-8853574 

•001035197 

967 

935089 

904231063 

31-0966236 

9-8887673 

•001034126 

968 

937024 

907039232 

31-1126984 

9-8921749 

•001033058 

969 

938961 

909853209 

31-1287648 

9-8955801 

•001031992 

970 

940900 

912673000 

31-1448230 

9-8989830 

•001030928 

971 

942841 

915498611 

31-1608729 

9-9023835 

•001029866 

972 

944784 

918330048 

31-1769145 

9-9057817 

•001028807 

973 

946729 

921167317 

31-1929479 

9-9091776 

•001027749 

974 

948676 

924010424 

31-2089731 

9-9125712 

•001026694 

975 

950625 

926859375 

31-2249900 

9-9159624 

-001025641 

976 

952576 

929714176 

31-2409987 

9-9193513 

•001024590 

977 

954529 

932574833 

31-2569992 

•9-9227379 

•001023541 

978 

956484 

935441352 

31-2729915 

9-9261222 

•001022495 

979 

958441 

938313739 

31-2889757 

9-9295042 

•001021450 

980 

960400 

941192000 

31-3049517 

9-9328839 

•001020408 

981 

962361 

944076141 

31-3209195 

9-9362613 

•001019168 

982 

964324 

946966168 

31-3368792 

9-9396363 

•001018330 

983 

966289 

949862087 

31-3528308 

9-9430092 

•001017294 

984 

968256 

952763904 

31-3687743 

9-9463797 

•001016260 

985 

970225 

955671625 

31-3847097 

9-9497479 

•001015228 

986 

972196 

958585256 

31-4006369 

9-9531138 

•001014199 

987 

974169 

961504803 

31-4165561 

9-9564775 

•001013171 

116 


THE   PRACTICAL   MODEL   CALCULATOR. 


Numbef. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

988 

976144 

964430272 

31-4324673 

9-9598389 

•001012146 

989 

978121 

967361669 

31-4483704 

9-9631981 

•001011122 

990 

980100 

970299000 

31-4642654 

9-9G65549 

•001010101 

991 

982081 

973242271 

31-4801525 

9-9699055 

•001009082 

992 

984064 

976191488 

31-4960315 

9-9732619 

•001008065 

993 

986049 

979146657 

31-5119025 

9-9766120 

•001007049 

994 

988036 

982107784 

31-5277655 

9-9799599 

-001006036 

995 

990025 

985074875 

31-5436206 

9-9833055 

•001005025 

996 

992016 

988047936 

31-5594677 

9-9866488 

•001004016 

997 

994009 

991026973 

31-5753068 

9-9899900 

O01003009 

998 

996004 

994011992 

31-5911380 

9-9933289 

-001002004 

999 

998001 

997002999 

31-6069613 

9-9966656 

•001001001 

1000 

1000000 

1000000000 

31-6227766 

10-0000000 

-001000000 

1001 

1000201 

1003003001 

31-6385840 

10-0033222 

-0009990010 

1002 

1004004 

1006012008 

31-6543866 

10-0066622 

•0009980040 

1003 

1006009 

1009027027 

31-6701752 

10-0099899 

•0009970090 

1004 

1008016 

1012048064 

31-6859590 

10-0133155 

•0009960159 

1005 

1010025 

1015075125 

31-7017349 

10-0166389 

•0009950249 

1006  - 

1010036 

1018108216 

31-7175030 

10-0199601 

•0009940358 

1007^ 

1014049 

1021147343 

31-7332633 

10-0232791 

•0009930487 

1008 

1016064 

1024192512 

31-7490157 

10-0265958 

•0009920635 

1009 

1018081 

1027243729 

31-7647603 

10-0299104 

•0009910803 

1010 

1020100 

1030301000 

31-7804972 

10-0332228 

-0009900990 

1011 

1020121 

1033364331 

31-7962262 

10-0365330 

•0009891197 

1012 

1024144 

1036433728 

31-8119474 

10-0398410 

•0009881423 

1013 

1026169 

1039509197 

31-8276609 

10-0431469 

•0009871668 

1014 

1028196 

1042590744 

31-8433666 

10-0464506 

•0009861933 

1015 

3030225 

1045678375 

31-8590646 

10-0497521 

•0009852217 

1016" 

1032256 

1048772096 

31-8747549 

10-0530514 

•0009842520 

1017 

1034289 

1051871913 

31-8904374 

10-0563485 

•0009832842 

1018 

1036324 

1054977832 

31-9061123 

10-0596435 

•0009823183 

1019 

1038361 

1058089859 

31-9217794 

10-0629364 

•0009813543 

1020 

1040400 

1061208000 

31-9374388 

10-0662271 

-0009803922 

1021 

1042441 

1064332261 

31-9530906 

10-0695156 

-0009794319 

1022 

1044484 

1067462648 

31-9687347 

10-0728020 

•0009784736 

1023 

1046529 

1070599167 

31-9843712 

10-0760863 

•0009775171 

1024 

1048576 

1073741824 

32-0000000 

10-0793684 

•0009765625 

1025 

1050625 

1076890625 

32-0156212 

10-0826484 

•0009756098 

1026 

1052676 

1080045576 

32-0312348 

10-0859262 

-0009746589 

1027 

1054729 

1083206683 

32-0468407 

10-0892019 

•0009737098 

1028 

1056784 

1086373952 

32-0624391 

10-0924755 

•0009727626 

1029 

1058841 

1089547389 

32-0780298 

10-0957469 

•0009718173 

1030 

1060900 

1092727000 

32-0936131 

10-0990163 

•0009708738 

1031 

1062961 

1095912791 

32-1091887 

10-1022835 

•0009699321 

1032 

1065024 

1099104768 

32-1247568 

10-1055487 

•0009689922 

1033 

1067089 

1102302937 

32-1403173 

10-1088117 

•0009680542 

1034 

1069156 

1105507304 

32-1558704 

10-1120726 

•0009671180 

1035 

1071225 

1108717875 

32-1714159 

10-1153314 

•0009661836 

1036 

1073296 

1111934656 

32-1869539 

10-1185882 

•0009652510 

1037 

1075369 

1115157653 

32-2024844 

10-1218428 

•0009643202 

1038 

1077444 

1118386872 

32-2180074 

10-1250953 

•0009633911 

1039 

1079521 

1121622319 

32-2335229 

10-1283457 

•0009624639 

1040 

1081600 

1124864000 

32-2490310 

10-1315941 

•0009615385 

1041 

1083681 

1128111921 

32-2645316 

10-1348403 

•0009606148 

1012 

1085764 

1131366088 

32-2800248 

10-1380845 

•0009596929 

1043 

1087849 

1134626507 

32-2955105 

10-1413266 

•0009587738 

1044 

1089936 

1137893184 

32-3109888 

10-1445667 

•0009578544 

1045 

1092025 

1141166125 

32-3264598 

10-1478047 

•0009569378 

1046 

1094116 

1144445336 

32-3419233 

10-1510406 

•0009560229 

1047 

1096209 

1147730823 

32-3573794 

10-1542744 

•0009551098 

1048 

1098304 

1151022592 

32-3728281 

10-1575062 

•0009541985 

1049 

1100401 

1154320049 

32-3882695 

10-1607359 

•0009532888 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    117 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1050 

1102500 

1157625000 

32-4037035 

10-1639636 

•0009523810 

1051 

1104601 

1160935651 

32-4191301 

10-1671893 

•0009514748 

1052 

1106704 

1164252608 

32-4345495 

10-1704129 

-0009505703 

1053 

1108809 

1167575877 

32-4499615 

10-1736344 

•0009496676 

1054 

1110916 

1170905464 

32-4653662 

10-1768539 

•0009487666 

1055 

1113125 

1174241375 

32-4807635 

10-1800714 

•0009478673 

1056 

1115136 

1177583616 

32-4961536 

10-1832868 

•0009469697 

1057 

1117249 

1180932193 

32-5115364 

10-1865002 

•0009460738 

1058 

1119364 

1184287112 

32-5269119 

10-1897116 

•0009451796 

1059 

1121481 

1187648379 

32-5422802 

10-1929209 

•0009442871 

1060 

1123600 

1191016000 

32-5576412 

10-1961283 

•0009433962 

1061 

1125721 

1194389981 

32-5729949 

10-1993336 

•0009425071 

1062 

1127844 

1197770328 

32-5883415 

10-2025369 

•0009416196 

1063 

1129969 

1201157047 

32-6035807 

10-2057382 

•0009407338 

1064 

1132096 

1204550144 

32-6190129 

10-2089375 

•0009398496 

1065 

1134225 

1207949625 

32-6343377 

10-2121347 

•0009389671 

1066 

1136356 

1211355496 

32-6496554 

10-2153300 

•0009380863 

1067 

1138489 

1214767763 

32-6649659 

10-2185233 

•0009372071 

1068 

1140624 

1218186432 

32-6802693 

10-2217146 

•0009363296 

1069 

1142761 

1221611509 

32-6955654 

10-2249039 

•0009354537 

1070 

1144900 

1225043000 

32-7108544 

10-2280912 

•0009345794 

1071 

1147041 

1228480911 

32-7261363 

10-2312766 

•0009337068 

1072 

1149184 

1231925248 

32-7414111 

10-2344599 

•0009328358 

1073 

1151329 

1235376017 

32-7566787 

10-2376413 

•0009319664 

1074 

1153476 

1238833224 

32-7719392 

10-2408207 

•0009310987 

1075 

1155625 

1242296875 

32-7871926 

10-2439981 

•0009302326 

1076 

1157776 

1245766976 

32-8024398 

10-2471735 

•0009293680 

1077 

1159929 

1249243533 

32-8176782 

10-2503470 

•0009285051 

1078 

1162084 

1252726552 

32-8329103 

10-2535186 

•0009276438 

1079 

1164241 

1256216039 

32-8481354 

10-2566881 

•0009267841 

1080 

1166400 

1259712000 

32-8633535 

10-2598557 

•0009259259 

1081 

1168561 

1263214441 

32-8785644 

10-2630213 

•0009250694 

1082 

1170724 

1266723368 

32-8937684 

10-2661850 

•0009242144 

1083 

1172889 

1270238787 

32-9089653 

10-2693467 

•0009233610 

1084 

1175056 

1273760704 

32-9241553 

10-2725065 

•0009225092 

1085 

1177225 

1277289125 

32-9393382 

10-2756644 

•0009216590 

1086 

1179396 

1280824056 

32-9545141 

10-2788203 

•0009208103 

1087 

1181569 

1284365503 

32-9696830 

10-2819743 

•0009199632 

1088 

1183744 

1287913472 

32-9848450 

10-2851264 

•0009191176 

1089 

1185921 

1291467969 

33-0000000 

10-2882765 

•0009182736 

1090 

1188100 

1295029000 

33-0151480 

10-2914247 

•0009174312 

1091 

1190281 

1298596571 

33-0302891 

10-2945709 

•0009165903 

1092 

1192464 

1302170688 

33-0454233 

19-2977153 

•0009157509 

1093 

1194649 

1305751357 

33-0605505 

10-3008577 

•0009149131 

1094 

1196836 

1309338584 

33-0756708 

10-3039982 

•0009140768 

1095 

1199025 

1512932375 

33-0907842 

10-3071368 

•0009132420 

1096 

1201216 

1316532736 

33-1058907 

10-3102735 

•0009124008 

1097 

1203409 

1320139673 

33-1209903 

10-3134083 

•0009115770 

1098 

1205604 

1323753192 

33-1360830 

10-3165411 

•0009107468 

1099 

1207801 

1327373299 

33-1511689 

10-3196721 

•0009099181 

1100 

1210000 

1331000000 

33-1662479 

10-3228012 

•0009090909 

1101 

1212201 

1334633301 

33-1813200 

10-3259284 

•0009082652 

1102 

1214404 

1338273208 

33-1963853 

10-3290537 

•0009074410 

1103 

1216609 

1341919727 

33-2114488 

10-3321770 

•0009066183 

1104 

1218816 

1345572864 

33-2266955 

10-3352985 

•0009057971 

1105 

1221025 

1349232625 

33-2415403 

10-3384181 

•0009049774 

1106 

1223236 

1352899816 

33-2565783 

10-3415358 

•0009041591 

1107 

1225449 

1356572043 

33-2716095 

10-3446517 

•0009033424 

1108 

1227664 

1360251712 

33-2866339 

10-3477657 

•0009025271 

1109 

1229881 

1363938029 

33-3016516 

10-3508778 

•0009017133 

1110 

1232100 

1367631000 

33-3166625 

10-3539880 

•0009009009 

1111 

1234321 

1371330631 

33-3316666 

10-3570964 

•0009000900 

118 


THE   PRACTICAL  MODEL   CALCULATOR, 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1112 

1236544 

1375036928 

33-3466640 

10-3602029 

•0008992806 

1113 

1238769 

1378749897 

33-3616546 

10-3033076 

•0008984726 

1114 

1240996 

1382469544 

33-3766385 

10-3664103 

•0008976661 

1115 

1243225 

1386195875 

33-3916157 

10-3695113 

•0008968610 

1116 

1245456 

1389928896 

33-4065862 

10-3726103 

•0008960753 

1117 

1247689 

1393668613 

33-4215499 

10-3757076 

•0008952551 

1118 

1249924 

1397415032 

33-4365070 

10-3788030 

-0008944544 

1119 

1252161 

1401168159 

33-4514573 

10-3818965 

•0008936550 

1120 

1254400 

1404928000 

33-4664011 

10-3849882 

•0008928571 

1121 

1256641 

1408694561 

33-4813381 

10-3880781 

•0008960607 

1122 

1258884 

1412467848 

33-4962684 

10-3911661 

•0008912656 

1123 

1261129 

1416247867 

33-5111921 

10-3942527 

-0008904720 

1124 

1263376 

1420034624 

33-5261092 

10-3973366 

•0008896797 

1125 

1265625 

1423828125 

33-5410196 

10-4004192 

•0008888889 

1126 

1267876 

1427628376 

33-5559234 

10-4034999 

•0008880995 

1127 

1270129 

1431435383 

33-5708206 

10-4065787 

•0008873114 

1128 

1272384 

1435249152 

33-5857112 

10-4096557 

•0008865248 

1129 

1274641 

1439069689 

33-6005952 

10-4127310 

•0008857396 

1130 

1276900 

1442897000 

33-6154726 

10-4158044 

•0008849558 

1131 

1279161 

1446731091 

33-6303434 

10-4188760 

•0008841733 

1132 

1281424 

1450571968 

33-6452077 

10-4219458 

•0008833922 

1133 

1283689 

1454419637 

33-6600653 

10-4250138 

-0008826125 

1134 

1285956 

1458274104 

33-6749165 

10-4280800 

•0008818342 

1135 

1288225 

1462135375 

33-6897610 

10-4311443 

•0008810573 

1136 

1290496 

1466003456 

33-7045991 

10-4342069 

•0008802817 

1137 

1292769 

1469878353 

33-7174306 

10-4372677 

•0008795075 

1138 

1295044 

1473760072 

33-7340556 

10-4403677 

•0008787346 

1139 

1297321 

1477648619 

33-7490741 

10-4433839 

•0008779631 

1140 

1299600 

1481544000 

33-7638860 

10-4464393 

•0008771930 

1141 

1301881 

1485446221 

33-7786915 

10-4494929 

•0008764242 

1142 

1304164 

1489355288 

33-7934905 

10-4525448 

•0008756567 

1143 

1306449 

1493271207 

33-8082830 

10-4555948 

•0008748906 

1144 

1308736 

1497193984 

33-8230691 

10-4586431 

•0008741259 

1145 

1311025 

1501123625 

33-8378486 

10-4616896 

•0008733624 

1146 

1313316 

1505060136 

33-8526218 

10-4647343 

•0008726003 

1147 

1315609 

1509003523 

33-8673884 

10-4677773 

•0008718396 

1148 

1317904 

1512953792 

33-8821487 

10-4708158 

•0008710801 

1149 

1320201 

1516910949 

33-8969025 

10-4738579 

-0008703220 

1150 

1322500 

1520875000 

33-9116499 

10-4768955 

•0008695652 

1151 

1324801 

1524845951 

33-9263909 

10-4799314 

•0008688097 

1152 

1327104 

1528823808 

33-9411255 

10-4829656 

•0008680556 

1153 

1329409 

1532808577 

33-9558537 

10-4859980 

•0008673027 

1154 

1331716 

1536800264 

33-9705755 

10-4890286 

•0008665511 

1155 

1334025 

1540798875 

33-9852910 

10-4920575 

•0008658009 

1156 

1336336 

1544804416 

34-0000000 

10-4950847 

•0008650519 

1157 

1338649 

1548816893 

34-0147027 

10-4981101 

•0008643042 

1158 

1340964 

1552836312 

34-0293990 

10-5011337 

•0008635579 

1159 

1343281 

1556862679 

34-0440890 

10-5041556 

-0008628128 

1160 

1345600 

1560896000 

34-0587727 

10-5071757 

•0008620690 

1161 

1347921 

1564936281 

34-0734501 

10-5101942 

•0008613244 

1162 

1350244 

1568983528 

34-0881211 

10-5132109 

•0008605852 

1163 

1352569 

1573037749 

34-0127858 

10-5162259 

•0008598452 

1164 

1354896 

1577098944 

34-1174442 

10-5192391 

•0008591065 

1165 

1357225 

1581167125 

34-1320963 

10-5222506 

•0008583691 

1166 

1359556 

1585242296 

34-1467422 

10-5252604 

•0008576329 

1167 

1361889 

1589324463 

34-1613817 

10-528268-3 

•0008568980 

1168 

1364224 

1593413632 

34-1760150 

10-5312749 

•0008561644 

1169 

1366561 

1597509809 

34-1906420 

10-5342795 

•0008554320 

1170 

1368900 

1601613000 

34-2052627 

10-5372825 

•0008547009 

1171 

1371241 

1605723211 

34-2198773 

10-5402837 

•0008539710 

1172 

1373584 

1609840448 

34-2344855 

10-5432832 

•0008532423 

1173 

1375929 

1613964717 

34-2490875 

10-5462810 

•0008525149 

TABLE  OF  SQUAKES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    119 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1174 

1378276 

1618096024 

34-2636834 

10-5492771 

•0008517888 

1175 

1380625 

1622234375 

34-2782730 

10-5522715 

•0008510638 

1176 

1382976 

1626379776 

34-2928564 

10-5552642 

•0008503401 

1177 

1385329 

1630532233 

34-3074336 

10-5582552 

•0008496177 

1178 

1387684 

1634691752 

34-3220046 

10-5612445 

•0008488964 

1179 

1390041 

1638858339 

34-3365694 

10-5642322 

•0008481764 

1180 

1392400 

1643032000 

34-3511281 

10-5672181 

•0008471576 

1181 

1394761 

1647212741 

34-3656805 

10-5702024 

•0008467401 

1182 

1397124 

1651400568 

34-3802268 

10-5731849 

•0008460237 

1183 

1399489 

1655595487 

34-3947670 

10-5761658 

•0008453085 

1184 

1401856 

1659797504 

34-4093011 

10-5791449 

•0008445946 

1185 

1404225 

1664006625 

34-4238289 

10-5821225 

•0008438819 

1186 

1406596 

1668222856 

34-4383507 

10-5850983 

•0008431703 

1187 

1408969 

1672446203 

34-4528663 

10-5880725 

•0008424600 

1188 

1411344 

1676676672 

34-4673759 

10-5910450 

•0008417508 

1189 

1413721 

1680914629 

34-4818793 

10-5940158 

•0008410429 

1190 

1416100 

1685159000 

34-4963766 

10-5969850 

•0008403361 

1191 

1418481 

1689410871 

34-5108678 

10-5999525 

•0008396306 

1192 

1420864 

1693669888 

34-5253530 

10-6029184 

•0008389262 

1193 

1423249 

1697936057 

34-5398321 

10-6058826 

•0008382320 

1194 

1425636 

1702209384 

34-5543051 

10-6088451 

•0008375209 

1195 

1428025 

1706489875 

34-5687720 

10-6118060 

•0008368201 

1196 

1430416 

1710777536 

34-5832329 

10-6147652 

•0008361204 

1197 

1432809 

1715072373 

34-5976879 

10-6177228 

•0008354219 

1198 

1435204 

1719374392 

34-6121366 

10-6206788 

•0008347245 

1199 

1437601 

1723683599 

34-6265794 

10-6236331 

•0008340284 

1200 

1440000 

1728000000 

34-6410162 

10-6265857 

•0008333333 

1201 

1442401 

1732323601 

34-6554469 

10-6295367 

•0008326395 

1202 

1444804 

1736654408 

34-6698716 

10-6324860 

•0008319408 

1203 

1447209 

1740992427 

34-6842904 

10-6354338 

•0008312552 

1204 

1449616 

1745337664 

34-6987031 

10-6383799 

•0008305648 

1205 

1452025 

1749690125 

34-7131099 

10-6413244 

•0008298755 

1206 

1454436 

1754049816 

34-7275107 

10-6442672 

•0008291874 

1207 

1456849 

1758416743 

34-7419055 

10-6472085 

•0008285004 

1208 

1459264 

1762790912 

34-7562944 

10-6501480 

•0008278146 

1209 

1461681 

1767172329 

34-7706773 

10-6530860 

•0008271299 

1210 

1464100 

1771561000 

34-7850543 

10-6560223 

•0008264463 

1211 

1466521 

1775956931 

34-7994253 

10-6589570 

•0008257638 

1212 

1468944 

1780360128 

34-8137904 

10-6618902 

•0008250825 

1213 

1471369 

1784770597 

34-8281495 

10-6648217 

•0008244023 

1214 

1473796 

1789188344 

34-8425028 

10-6677516 

•0008237232 

1215 

1476225 

1793613375 

34-8568501 

10-6706799 

•0008230453 

1216 

1478656 

1798045696 

34-8711915 

10-6736066 

•0008223684 

1217 

1481089 

1802485313 

34-8855271 

10-6765317 

•0008216927 

1218 

1483524 

1806932232 

34-8998567 

10-6794552 

•0008210181 

1219 

1485961 

1811386459 

34-9141805 

10-6823771 

•0008203445 

1220 

1488400 

1815848000 

34-9284984 

10-6852973 

•0008196721 

1221 

1490841 

1820316861 

34-9428104 

10-6882160 

•0008190008 

1222 

1493284 

1824793048 

34-9571166 

10-6911331 

•0008183306 

1223 

1495729 

1829276567 

34-9714169 

10-6940486 

•0008176615 

1224 

1498176 

1833764247 

34-9857114 

10-6969625 

•0008169935 

1225 

1500625 

1838265625 

35-0000000 

10-6998748 

•0008163265 

1226 

1503276 

1842771176 

35-0142828 

10-7027855 

•0008156607 

1227 

1505529 

1847284083 

35-0285598 

10-7056947 

•0008149959 

1228 

1507984 

1851804352 

35-0428309 

10-7086023 

•0008143322 

1229 

1510441 

1856331989 

35-0570963 

10-7115083 

•0008136696 

1230 

1512900 

1860867000 

35-0713558 

10-7144127 

•0008130081 

1231 

1515361 

1865409391 

35-0856096 

10-7173155 

•0008123477 

1232 

1517824 

1869959168 

35-0998575 

10-7202168 

•0008116883 

1233 

1520289 

1874516337 

35-1140997 

10-7231165 

•0008110300 

1234 

1522756 

1879080904 

35-1283361 

10-7260146 

•0008103728 

1235 

1525225 

1883652875 

35-1425568 

10-7289112 

•0008097166 

120 


THE    PRACTICAL   MODEL    CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1236 

1527696 

1888232256 

35-1567917 

10-7318062 

•0008090615 

1237 

1530169 

1892819053 

35-1710108 

10-7346997 

•0008084074 

1238 

1532644 

1897413272 

35-1852242 

10-7375916 

•0008077544 

1239 

1535121 

1902014919 

35-1994318 

10-7404819 

•0008071025 

1240 

1537600 

1906624000 

35-2136337 

10-7433707 

-0008064516 

1241 

1540081 

1911240521 

35-2278299 

10-7462579 

•0008058018 

1242 

1542564 

1915864488 

35-2420204 

10-7491436 

•0008051530 

1243 

1545049 

1920495907 

35-2562051 

10-7520277 

•0008045052 

1244 

1547536 

1925134784 

35-2703842 

10-7549103 

•0008038585 

1245 

1550025 

1929781125 

35-2845575 

10-7577913 

•0008032129 

1246 

1552521 

1934434936 

35-2987252 

10-7606708 

•0008025682 

1247- 

1555009 

1939096223 

35-3128872 

10-7635488 

•0008019246 

1248 

1557504 

1943764992 

35-3270435 

10-7664252 

•0008012821 

1249 

1560001 

1948441249 

35-3411941 

10-7693001 

•0008006405 

1250 

1562500 

1953125000 

35-3553391 

10-7721735 

•0008000000 

1251 

1565001 

1957816251 

35-3694784 

10-7750453 

•0007993605 

1252 

1567504 

1962515008 

35-3836120 

10-7779156 

•0007987220 

1253 

1570009 

1967221277 

35-3977400 

10-7807843 

•0007980846 

1254 

1572516 

1971935064 

35-4118624 

10-7836516 

•0007974482 

1255 

1575025 

1976656375 

35-4259792 

10-7865173 

•0007968127 

1256 

1577536 

1981385216 

35-4400903 

10-7893815 

•0007961783 

1257 

1580049 

1986121593 

35-4541958 

10-7922441 

•0007955449 

1258 

1582564 

1990865512 

354682957 

10-7951053 

•0007949126 

1259 

1585081 

1995616979 

35-4823900 

10-7979649 

•0007942812 

1200 

1587600 

2000376000 

35-4964787 

10-8008280 

•0007936508 

12G1 

1590121 

2005142581 

35-5105618 

10-8036797 

•0007930214 

1262 

1592644 

2009916728 

35-5246393 

10-8065348 

•0007923930 

1263 

1595166 

2014698447 

35-5387113 

10-8093884 

•0007917656 

1264 

1597696 

2019487744 

35-5527777 

10-8122404 

•0007911392 

1265 

1600225 

20242^4625 

35-5668385 

10-8150909 

•0007905138 

1266 

1602756 

2029089096 

35-5808937 

10-8179400 

•0007898894 

1207 

1605289 

2033901163 

35-5949434 

10-8207876 

•0007892660 

1268 

1607824 

2038720832 

35-6089876 

10-8236336 

•0007886435 

1269 

1610361 

2043548109 

35-6230262 

10-8264782 

•0007880221 

1270 

1612900 

2048383000 

35-6370593 

10-8293213 

•0007874016 

1271 

1615441 

2053225511 

35-6510869 

10-8321629 

•0007867821 

1272 

1617984 

2058075648 

35-6651090 

10-8350030 

•0007861635 

1273 

1620529 

2062933417 

35-6791255 

10-8378416 

•0007855460 

1274 

1623076 

2067798824 

35-6931366 

10-8406788 

•0007849294 

1275 

1625625 

2072671875 

35-7071421 

10-8435144 

•0007843137 

1276 

1628176 

2077552576 

35-7211422 

10-8463485 

•0007836991 

1277 

1630729 

2082440933 

35-7351367 

10-8491812 

•0007830854 

1278 

1633284 

2087336952 

35-7491258 

10-8520125 

•0007824726 

1279 

1635841 

'  2092240639 

35-7631095 

10-8548422 

•0007818608 

1280 

1638400 

2097152000 

35-7770876 

10-8576704 

•0007812500 

1281 

1640961 

2102071841 

35-7910603 

10-8604972 

•0007806401 

1282 

1643524 

2106997768 

35-8050276 

10-8633225 

•0007800312 

1283 

1646089 

2111932187 

35-8189894 

10-8661454 

•0007794232 

1284 

1648656 

2116874304 

35-8329457 

10-8689687 

•0007788162 

1285 

1651225 

2121824125 

35-8468966 

10-8717897 

•0007782101 

1286 

1653796 

2126781656 

35-8608421 

10-8746091 

•0007776050 

1287 

1656369 

2131746903 

35-8747822 

10-8774271 

•0007770008 

1288 

1658944 

2136719872 

35-8887169 

10-8802436 

•0007763975 

1289 

1661521 

2141700569 

35-9026461 

10-8830587 

•0007757952 

1290 

1664100 

2146689000 

35-9165699 

10-8858723 

•0007751938 

1291 

1666681 

2151685171 

35-9304884 

10-8886845 

•0007745933 

1292 

1669264 

2156689088 

35-9444015 

10-8914952 

•0007739938 

1293 

1671849 

2161700757 

35-9583092 

10-8943044 

•0007733952 

1294 

1674436 

2166720184 

35-9722115 

10-8971123 

•0007727975 

1295 

1677025 

2171747375 

35-9861084 

10-8999186 

•0007722008 

1296 

1679616 

2176782336 

36-0000000 

10-9027235 

•0007710049 

1297 

1682209 

2181825073 

36-0138862 

10-9055269 

•0007710100 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    121 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1298 

1684804 

2186875592 

36-0277671 

10-9083290 

•0007704160 

1299 

1687401 

2191933899 

36-0416426 

10-9111296 

•0007698229 

1300 

1690000 

2197000000 

36-0555128 

10-9139287 

•0007692308 

1301 

1692601 

2202073901 

36-0693776 

10-9167265 

•0007686395 

1302 

1695204 

2207155608 

36-0832371 

10-9195228 

•0007680492 

1303 

1697809 

2212245127 

36-0970913 

10-9223177 

•0007674579 

1304 

1700416 

2217342464 

36-1109402 

10-9251111 

•0007668712 

1305 

1703025 

2222447625 

36-1247837 

10-9279031 

•0007662835 

1306 

1705636 

2227560616 

36-1386220 

10-9306937 

•0007656968 

1307 

1708249 

2232681443 

36-1524550 

10-9334829 

•0007651109 

1308 

1710864 

2237810112 

36-1662826 

10-9362706 

•0007645260 

1309 

1713481 

2242946629 

36-1801050 

10-9390569 

•0007639419 

1310 

1716100 

2248091000 

36-1939221 

10-9418418 

•0007633588 

1311 

1718721 

2253243231 

36-2077340 

10-9446253 

•0007627765 

1312 

1721344 

2258403328 

36-2215406 

10-9475074 

•0007621951 

1313 

1723969 

2263571297 

36-2353419 

10-9501880 

•0007616446 

1314 

1726596 

2268747144 

36-2491379 

10-9529673 

•0007610350 

1315 

1729225 

2273930875 

36-2626287 

10-9557451 

•0007604563 

1316 

1731856 

2279122496 

36-2767143 

10-9585215 

•0007598784 

1317 

1734489 

2284322013 

36-2904246 

10-9612965 

•0007593014 

1318 

1737124 

2289529432 

36-3042697 

10-9640701 

•0007587253 

1319 

1739761 

2294744759 

36-3180396 

10-9668423 

•0007581501 

1320 

1742400 

2299968000 

36-3318042 

10-9696131 

•0007575758 

1321 

1745041 

2305199161 

36-3455637 

10-.9723825 

•0007570023 

1322 

1747684 

2310438248 

36-3593179 

10-9751505 

•0007564297 

1323 

1750329 

2315685267 

36-3730670 

10-9779171 

•0007558579 

1324 

1752976 

2320940224 

36-3868108 

10-9806823 

•0007552870 

1325 

1755625 

2326203125 

36-4005494 

10-9834462 

•0007547170 

1326 

1758276 

2331473976 

36-4142829 

10-9862086 

•0007541478 

1327 

1760929 

2336752783 

36-4280112 

10-9889696 

•0007535795 

1328 

1763584 

2342039552 

36-4417343 

10-9917293 

•0007530120 

1329 

1766241 

2347334289 

36-4554523 

10-9944876 

•0007524454 

1330 

1768900 

2352637000 

36-4691650 

10-9972445 

•0007518797 

1331 

1771561 

2357947691 

36-4828727 

11-0000000 

•0007513148 

1332 

1774224 

2363266368 

36-4965752 

11-0027541 

•0007507508 

1333 

1776889 

2368593037 

36-5102725 

11-0055069 

•0007501875 

1334 

1779556 

2373927704 

36-5239647 

11-0082583 

•0007496252 

1335 

1782225 

2379270375 

36-5376518 

11-0110082 

•0007490637 

1336 

1784896 

2384621056 

36-5513888 

11-0137569 

•0007485030 

1337 

1787569 

2389979753 

36-5650106 

11-0165041 

•0007479432 

1338 

1790244 

2395346472 

36-5786823 

11-0192500 

•0007473842 

1339 

1792921 

2400721219 

36-5923489 

11-0219945 

•0007468260 

1340 

1795600 

2406104000 

36-6060104 

11-0247377 

•0007462687 

1341 

1798281 

2411494821 

36-6196668 

11-0274795 

•0007457122 

1342 

1800964 

2416893688 

36-6333181 

11-0302199 

•0007451565 

1343 

1803649 

2422300607 

36-6469144 

11-0329590 

•0007446016 

1344 

1806336 

2427715584 

36-6606056 

11-0356967 

•0007440476 

1345 

1809025 

2433138625 

36-6742416 

11-0384330 

•0007434944 

1346 

1811716 

2438569736 

36-6878726 

11-0411680 

•0007429421 

1347 

1814409 

2444008923 

36-7014986 

11-0439017 

•0007423905 

1348 

1817104 

2449456192 

36-7151195 

11-0466339 

•0007418398 

1349 

1819801 

2454911549 

36-7287353 

11-0493649 

•0007412898 

1350 

1822500 

2460375000 

36-7423461 

11-0520945 

•0007407407 

1351 

1825201 

2465846551 

36-7559519 

11-0548227 

•0007401924 

1352 

1827904 

2471326208 

36-7695526 

11-0575497 

•0007396450 

1353 

1830609 

2476813977 

36-7831483 

11-0602752 

•0007390983 

1354 

1833316 

2482309864 

36-7967390 

11-0629994 

•0007385524 

1355 

1836025 

2487813875 

36-8103246 

11-0657222 

•0007380074 

1356 

1838736 

2493326016 

36-8239053 

11-0684437 

•0007374631 

1357 

1841449 

2498846293 

36-8374809 

11-0711639 

•0007369197 

1358 

1844164 

2504374712 

36-8510515 

11-0738828 

•0007363770 

1359 

1846881 

2509911279 

36-8646172 

11-0766003 

•0007358352 

122 


THE   PRACTICAL   MODEL   CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1360 

1849600 

2515456000 

36-8781778 

11-0793165 

•0007352941 

1361 

1852321 

2521008881 

36-8917335 

11-0820314 

•0007347539 

1362 

1855044 

2526569928 

36-9052842 

11-0847449 

•0007342144 

1363 

1857769 

2532139147 

36-9188299 

11-0874571 

•0007336757 

1364 

1860496 

2537716544 

36-9323706 

11-0901679 

•0007331378 

1365 

1863225 

2543302125 

36-9459064 

11-0928775 

•0007326007 

1366 

1865956 

2548895896 

36-9594372 

11-0955857 

•0007320644 

1367 

1868689 

2554497863 

36-9729631 

11-0982926 

•0007315289 

1368 

1871424 

2560108032 

36-9864840 

11-1009982 

•0007309942 

1369 

1874161 

2565726409 

37-0000000 

11-1037025 

•0007304602 

1370 

1876900 

2571353000 

37-0135110 

11-1064054 

•0007299270 

1371 

1879641 

2576987811 

37-0270172 

11-1091070 

•0007293946 

1372 

1882384 

2582630848 

37-0405184 

11-1118073 

•0007288630 

1373 

1885129 

2588282117 

37-0540146 

11-1145064 

•0007283321 

1374 

1887876 

2593941624 

37-0675060 

11-1172041 

•0007278020 

1375 

1890625 

2599609375 

37-0899924 

11-1199004 

•0007272727 

1376 

1893376 

2605285376 

37-0944740 

11-1225955 

•0007267442 

1377 

1896129 

2610969633 

37-1079506 

11-1252893 

•0007262164 

1378 

1898884 

2616662152 

37-1214224 

11-1279817 

•0007256894 

1379 

1901641 

2622362939 

37-1348893 

11-1306729 

•0007251632 

1380 

1904400 

2628072000 

37-1483512 

11-1333628 

•0007246377 

1381 

1907161 

2633789341 

37-1618084 

11-1360514 

•0007241130 

1382 

1909924 

2639514968 

37-1752606 

11-1387386 

•0007235890 

1383 

1912689 

2645248887 

37-1887079 

11-1414246 

•0007230658 

1384 

1915456 

2650991104 

37-2021505 

11-1441093 

•0007225434 

1385 

1918225 

2656741625 

37-2155881 

11-1467926 

•0007220217 

1386 

1920996 

2662500456 

37-2290209 

11-1494747 

•0007215007 

1387 

1923769 

2668267603 

37-2424489 

11-1521555 

•0007209805 

1388 

1926544 

2674043072 

37-2558720 

11-1548350 

•0007204611 

1389 

1929321 

2679826869 

37-2692903 

11-1575133 

•0007199424 

1390 

1932100 

2685619000 

37-2827037 

11-1601903 

•0007194245 

1391 

1934881 

2691419471 

37-2961124 

11-1628659 

•0007189073 

1392 

1937664 

2697228288 

37-3095162 

11-1655403 

•0007183908 

1393 

1940449 

2703045457 

37-3229152 

11-1682134 

•0007178751 

1394 

1943236 

2708870984 

37-3363094 

11-1708852 

•0007173601 

1395 

1946025 

2714704875 

37-3496988 

11-1735558 

•0007168459 

1396 

1948816 

2720547136 

37-3630834 

11-1762250 

•0007163324 

1397 

1951609 

2726397773 

37-3764632 

11-1788930 

•0007158196 

1398 

1954404 

2732256792 

37-3898382 

11-1815598 

•0007153076 

1399 

1957201 

2738124199 

37-4032084 

11-1842252 

•0007147963 

1400 

1960000 

2744000000 

37-4165738 

11-1868894 

•0007142857 

1401 

1962801 

2749884201 

37-4299345 

11-1895523 

•0007137759 

1402 

1965604 

2755776808 

37-4432904 

11-1922139 

•0007132668 

1403 

1968409 

2761677827 

37-4566416 

11-1948743 

•0007127584 

1404 

1971216 

2767587264 

37-4699880 

11-1975334 

•0007122507 

1405 

1974025 

2773505123 

37-4833296 

11-2001913 

•0007117438 

1406 

1976836 

2779431416 

37-4966665 

11-2028479 

•0007112376 

1407 

1979649 

2785366143 

37-5099987 

11-2055032 

•0007107321 

1408 

1982464 

2791309312 

37-5233261 

11-2081573 

•0007102273 

1409 

1985281 

2797260929 

37-5366487 

11-2108101 

•0007097232 

1410 

1988100 

2803221000 

37-5499667 

11-2134617 

•0007092199 

1411 

1990921 

2809189531 

37-5632799 

11-2161120 

•0007087172 

1412 

1993744 

2815166528 

37-5765885 

11-2187611 

•0007082153 

1413 

1996569 

2821151997 

37-5898922 

11-2214089 

•0007077141 

1414 

1999396 

2827145944 

37-6031913 

11-2240054 

•0007072136 

1415 

2002225 

2833148375 

37-6164857 

11-2267007 

•0007067138 

1416 

2005056 

2839159296 

37-6297754 

11-2293448 

•0007062147 

1417 

2007889 

2845178713 

37-6430604 

11-2319876 

•0007057163 

1418 

2010724 

2851206632 

37-6563407 

11-2346292 

•0007052186 

1419 

2013561 

2857243059 

37-6696164 

11-2372696 

•0007047216 

1420 

2016400 

2863288000 

37-6828874 

11-2399087 

•0007042254 

1421 

2019241 

2869341461 

37-6961536 

11-2425465 

•0007037298 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS.    123 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1422 

2022084 

2875403448 

37-7094153 

11-2451831 

•0007032349 

1423 

2024929 

2881473967 

37-7226722 

11-2478185 

•0007027407 

1424 

2027776 

2887553024 

37-7359245 

11-2504527 

•0007022472 

1425 

2030625 

2893640625 

37-7491722 

11-2530856 

•0007017544 

1426 

2033476 

2899736776 

37-7624152 

11-2557173 

•0007012623 

1427 

2036329 

2905841483 

37-7756535 

11-2583478 

•0007007708 

1428 

2039184 

2911954752 

37-7888873 

11-2609770 

•0007002801 

1429 

2042041 

2918076589 

37-8021163 

11-2636050 

•0006997901 

1430 

2044900 

2924207000 

37-8153408 

11-2662318 

•0006993007 

1431 

2047761 

2930345991 

37-8285606 

11-2688573 

•0006988120 

1432 

2050624 

2936493568 

37-8417759 

11-2714816 

•0006983240 

1433 

2053489 

2942649737 

37-8549864 

11-2741047 

•0006978367 

1434 

2056356 

2948814504 

37-8681924 

11-2767266 

•0006973501 

1435 

2059225 

2954987875 

37-8813938 

11-2793472 

•0006968641 

1436 

2062096 

2961169856 

37-8945906 

11-2819666 

•0006963788 

1437 

2064969 

2967360453 

37-9077828 

11-2845849 

•0006958942 

1438 

2067844 

2973559672 

37-9209704 

11-2872019 

•0006954103 

1439 

2070721 

2979767519 

37-9341538 

11-2898177 

•0006949270 

1440 

2073600 

2985984000 

37-9473319 

11-2924323 

•0006944444 

1441 

2076481 

2992209121 

37-9605058 

11-2950457 

•0006939625 

1442 

2079364 

3098442888 

37-9736751 

11-2976579 

•0006934813 

1443 

2082249 

3004685307 

37-9868398 

11-3002688 

•0006930007 

1444 

2085136 

3010936384 

38-0000000 

11-3028786 

•0006925208 

1445 

2088025 

3017196125 

38-0131556 

11-3054871 

•0006920415 

1446 

2080916 

3023464536 

38-0263067 

11-3080945 

•0006915629 

1447 

2093809 

3029741623 

38-0394532 

11-3107006 

•0006910850 

1448 

2096704 

3036027392 

38-0525952 

11-3133056 

•0006906078 

1449 

2099601 

3042321849 

38-0657326 

11-3159094 

•0006901312 

1450 

2102500 

3048625000 

38-0788655 

11-3185119 

•0006896552 

1451 

2105401 

3054936851 

38-0919939 

11-3211132 

•0006891799 

1452 

2108304 

3061257408 

38-1051178 

11-3237134 

•0006887052 

1453 

2111209 

3067586777 

38-1182371 

11-3263124 

•0006882312 

1454 

2114116 

3073924664 

38-1313519 

11-3289102 

•0006877579 

1455 

2117025 

3080271375 

38-1444622 

11-3315067 

•0006872852 

1456 

2119936 

3086626816 

38-1575681 

11-3341022 

•0006868132 

1457 

2122849 

3092990993 

38-1706693 

11-3366964 

•0006863412 

1458 

2125764 

3099363912 

38-1837662 

11-3392894 

•0006858711 

1459 

2128681 

3105745579 

38-1968585 

11-3418813 

•0006854010 

1460 

2131600 

3112136000 

38-2099463 

11-3444719 

•0006849315 

1461 

2134521 

3118535181 

38-2230297 

11-3470614 

•0006844627 

1462 

2137444 

3124943128 

38-2361085 

11-3496497 

•0006839945 

1463 

2140369 

3131359847 

38-2491829 

11-3522368 

•0006835270 

1464 

2143296 

3137785344 

38-2622529 

11-3548227 

•0006830601 

1465 

2146225 

3144219625 

38-2753184 

11-3574075 

•0006825939 

1466 

2149156 

3150662696 

38-2883794 

11-3599911 

•0006821282 

1467- 

2152089 

3157114563 

38-3014360 

11-3625735 

•0006816633 

1468 

2155024 

3163575232 

38-3144881 

11-3651547 

•0006811989 

1469 

2157961 

3170044709 

38-3275358 

11-3677347 

•0006807352 

1470 

2160900 

3176523000 

38-3405790 

11-3703136 

•0006802721 

1471 

2163841 

3183010111 

88-3536178 

11-3728914 

•0006798097 

1472 

2166784 

3189506048 

38-3666522 

11-3754679 

•0006793478 

1473 

2169729 

3196010817 

38-3796821 

11-3780433 

•0006788866 

1474 

2172676 

3202524424 

38-3927076 

11-3806175 

•0006784261 

1475 

2175625 

3209046875 

38-4057287 

11-3831906 

•0006779661 

1476 

2178576 

3215578176 

38-4187454 

11-3857625 

•0006775068 

1477 

2181529 

3222118333 

38-4317577 

11-3883332 

•0006770481 

1478 

2184484 

3228667352 

38-4447656 

11-3909028 

•0006765900 

1479 

2187441 

3235225239 

38-4577691 

11-3934712 

•0006761325 

1480 

2190400 

3241792000 

38-4707681 

11-3960384 

•0006756757 

1481 

2193361 

3248367641 

38-4837627 

11-3986045 

•0006752194 

1482 

2196324 

3254952168 

38-4967530 

11-4011695 

•0006747638 

1483 

2199289 

3261545587 

38-5097390 

11-4037332 

•0006743088 

124 


THE   PRACTICAL   MODEL   CALCULATOR. 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1484 

2202256 

3268147904 

38-5227206 

11-4062959 

•0006738544 

1485 

2205225 

3274759125 

38-5356977 

11-4088574 

•0006784007 

1486 

2208196 

3281379256 

38-5486705 

11-4114177 

•0006729474 

1487 

2211169 

3288008303 

38-5616389 

11-4139769 

•0006724950 

1488 

2214144 

3294646272 

38-5746030 

11-4165349 

•0006720430 

1489 

2217121 

3301293169 

38-5875627 

11-4190918 

•0006715917 

1490 

2220100 

3307949000 

38-6005181 

11-4206476 

•0006711409 

1491 

2223081 

3314613771 

38-6134691 

11-4242022 

•0006706908 

1492 

2226004 

3321287488 

38-6264158 

11-4267556 

•0006702413 

1493 

2229049 

3227970157 

38-6393582 

11-4293079 

•0006697924 

1494 

2232036 

3334661784 

38-6522962 

11-4318591 

•0006693440 

1495 

2235025 

3341362375 

38-6652299 

11-4344092 

•0006688963 

1496 

2238016 

3348071936 

38-6781593 

11-4369581 

•0006684492 

1497 

2241009 

3354790473 

38-6910843 

11-4395059 

•0006680027 

1498 

2244004 

3361517992 

38-7040050 

11-4420525 

•0006675567 

1499 

2247001 

3368254499 

38-7169214 

11-4445980 

•0006671114 

1500 

2250000 

3375000000 

38-7298335 

11-4471424 

•0006666667 

1501 

2253001 

3381754501 

38-7427412 

11-4496857 

•0006662225 

1502 

2256004 

3388518008 

38-7556447 

11-4522278 

•0006657790 

1503 

2259009 

3395290527 

38-7685439 

11-4547688 

•0006553360 

1504 

2262016 

3402072064 

38-7814389 

11-4573087 

•0006648936 

1505 

2265025 

3408862625 

38-7943294 

11-4598476 

•0006644518 

1506 

2268036 

3415662216 

38-8072158 

11-4623850 

•0006640106 

1507 

2271049 

3422470843 

38-8200978 

11-4649215 

•0006635700 

1508 

2274064 

3429288512 

38-8329757 

11-4674568 

•0006631300 

1509 

2277081 

3436115229 

38-8458491 

11-4699911 

•0006626905 

1510 

2280100 

3442951000 

38-8587184 

11-4725242 

•0006622517 

1511 

2283121 

3449795831 

38-8715834 

11-4750562 

•0006618134 

1512 

2286144 

3456649728 

38-8844442 

11-4775871 

•0006613757 

1513 

2289169 

3463512697 

38-8973006 

11-4801169 

•0006609385 

1514 

2292196 

3470384744 

38-9101529 

11-4826455 

•0006605020 

1515 

2295225 

3477265875 

38-9230009 

11-4851731 

•0006600660 

1516 

2298256 

3484156096 

38-9358447 

11-4876995 

•0006596306 

1517 

2301289 

3491055413 

38-9486841 

11-4902249 

•0006591958 

1518 

2304324 

3597963832 

38-9615194 

11-4927491 

•0006587615 

1519 

2307361 

3504881359 

38-9743505 

11-4952722 

•0006583278 

1520 

2310400 

3511808000 

38-9871774 

11-4977942 

•0006578947 

1521 

2313441 

3518743761 

39-0000000 

11-5003151 

•0006574622 

1522 

2316484 

3525688648 

39-0128184 

11-5028348 

•0006570302 

1523 

2319529 

3532642667 

39-0256326 

11-5053535 

•0006565988 

1524 

2322576 

3539605824 

39-0384426 

11-5078711 

•0006561680 

1525 

2325625 

3546578125 

39-0512483 

11-5103876 

•0006557377 

1526 

2328676 

3553559576 

39-0640499 

11-5129030 

•0006553080 

1527 

2331729 

3567549552 

39-0768473 

11-5154173 

•0006548788 

1528 

2334784 

3560558183 

39-0896406 

11-5179305 

•0006544503 

1529 

2337841 

3574558889 

39-1024296 

11-5204425 

•0006540222 

1530 

2340900 

3581577000 

39-1152144 

11-5229535 

•0006535948 

1531 

2343961 

3588604291 

39-1279951 

11-5254634 

•0006531679 

1532 

2347024 

3595640768 

39-1407716 

11-5279722 

•0006527415 

1533 

2350089 

3602686437 

39-1535439 

11-5304799 

•0006523157 

1534 

2353156 

3609741304 

39-1663120 

11-5329865 

•0006518905 

1535 

2356225 

3616805375 

39-1790760 

11-5354920 

•0000514658 

1536 

2359256 

3623878656 

39-1918359 

11-5379965 

•0006510417 

1537 

2362369 

3630961153 

39-2045915 

11-5404998 

•0006506181 

1538 

2365444 

3638052872 

39-2173431 

11-5430021 

•0006501951 

1539 

2368521 

3645153819 

39-2300905 

11-5455033 

•0006497726 

1540 

2371600 

3652264000 

39-2428337 

11-5480034 

•0006493506 

1541 

2374681 

3659383421 

39-2555728 

11-5505025 

•0006489293 

1542 

2377764 

3666512088 

39-2683078 

11-5530004 

•0006485084 

1543 

2380849 

3673650007 

39-2810387 

11-5554972 

•0006480881 

1544 

2383936 

3680797184 

39-2937654 

11-5579931 

•0006476684 

1545 

2387025 

3687953625 

39-3064880 

11-5604878 

•0006472492 

TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS. 


125 


Number. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

154(3 

2390116 

3695119336 

39-3192065 

11-5629815 

•0006468305 

1547 

2393209 

3702294323 

39-3319208 

11-5654740 

•0006464124 

1548 

2396304 

3709478592 

39-3446311 

11-5679655 

•0006459948 

1549 

2399401 

3716672149 

39-3573373 

11-5704559 

•0006455778 

1550 

2402500 

3723875000 

39-3700394 

11-5729453 

•0006451613 

1551 

2405601 

3731087151 

39-3827373 

11-5754336 

•0006447453 

1552 

2408704 

3738308608 

39-3954312 

11-5779208 

•0006443299 

1553 

2411809 

3745539377 

39-4081210 

11-5804069 

•0006439150 

155i 

2414916 

3752779464 

39-4208067 

11-5828919 

•0006435006 

1555 

2418025 

3760028875 

39-4334883 

11-5853759 

•0006430868 

1556 

2421136 

3767287616 

39-4461658 

11-5878588 

•0006426735 

1557 

2424249 

3774555693 

39-4588393 

11-5903407 

•0006422608 

1558 

2427364 

3781833112 

39-4715087 

11-5928215 

•0006418485 

1559 

2430481 

3789119879 

39-4841740 

11-5953013 

•0006414368 

1560 

2433600 

3796416000 

39-4968353 

11-5977799 

•0006410256 

1561 

2436721 

3803721481 

39-5094925 

11-6002576 

•0006406150 

1562 

2439844 

3811036328 

39-5221457 

11-6027342 

•0006402049 

1563 

2442969 

3818360547 

39-5347948 

11-6052097 

•0006397953 

1564 

2446096 

3825641444 

39-5474399 

11-6076841 

•0006393862 

1565 

2449225 

3833037125 

39-5600809 

11-6101575 

•0006389776 

1566 

2452356 

3840389496 

39-5727179 

11-6126299 

•0006385696 

1567 

2455489 

3847751263 

39-5853508 

11-6151012 

•0006381621 

1568 

2458624 

3855123432 

39-5979797 

11-6175715 

•0006377551 

1569 

2461761 

3862503009 

39-6106046 

11-6200407 

•0006373486 

1570 

2464900 

3869883000 

39-6232255 

11-6225088 

•0006369427 

1571 

2468041 

3877292411 

39-6358424 

11-6249759 

•0006365372 

1572 

2471184 

3884701248 

39-6484552 

11-6274420 

•0006361323 

1573 

2474329 

3892119157 

39-6610640 

11-6299070 

•0006357279 

1574 

2477476 

3899547224 

39-6736688 

11-6323710 

•0006353240 

1575 

2480625 

3906984375 

39-6862696 

11-6348339 

•0006349206 

1576 

2483776 

3914430976 

39-6988665 

11-6372957 

•0006345178 

1577 

2486929 

3921887033 

39-7114593 

11-6397566 

•0006341154 

1578 

2490084 

3929352552 

39-7240481 

11-6422164 

•0006337136 

1579 

2493241 

3936827539 

39-7366329 

11-6446751 

•0006333122 

1580 

2496400 

3944312000 

39-7492138 

11-6471329 

•0006329114 

1581 

2499561 

3951805941 

39-7617907 

11-6495895 

•0006325111 

1582 

2502724 

3959309368 

39-7743636 

11-6520452 

•0006321113 

1583 

2505889 

3966822287 

39-7869325 

11-6544998 

•0006317119 

1584 

2509056 

3974344704 

39-7994976 

11-6569534 

•0006313131 

1585 

2512225 

3981876625 

39-8120585 

11-6594059 

•0006309148 

1586 

2515396 

3989418056 

39-8246155 

11-6618574 

•0006305170 

1587 

2518569 

3996969003 

39-8371686 

11-6643079 

•0006301197 

1588 

2521744 

4004529472 

39-8497177 

11-6667574 

•0006297229 

1589 

2524921 

4012099469 

39-8622628 

11-6692058 

•0006293266 

1590 

2528100 

4014679000 

39-8748040 

11-6716532 

•0006289308 

1591 

2531281 

4027268071 

39-8873413 

11-6740996 

•0006285355 

1592 

2534464 

4034866688 

39-8998747 

11-6765449 

•0006281407 

1593 

2537649 

4042474857 

39-9124041 

11-6789892 

•0006277464 

1594 

2540836 

4050092584 

39-9249295 

11-6814325 

•0006273526 

1595 

2544025 

4057719875 

89-9374511 

11-6838748 

•0006269592 

1596 

2547216 

4065356736 

39-9499687 

11-6863161 

•0006265664 

1597 

2550409 

4073003173 

39-9624824 

11-6887563 

•0006261741 

1598 

2553604 

4080659192 

39-9749922 

11-6911955 

•0006257822 

1599 

2556801 

4088324799 

39-9874980 

11-6936337 

•0006253909 

1600 

2560000 

4096000000 

40-0000000 

11-6960709 

•0006250000 

To  find  the  square  or  cube  root  of  a  number  consisting  of  integers 

and  decimals. 

RULE. — Multiply  the  difference  between  the  root  of  the  integer 
part  of  the  given  number,  and  the  root  of  the  next  higher  integer 
number,  by  the  decimal  part  of  the  given  number,  and  add  tho 


126          THE  PRACTICAL  MODEL  CALCULATOR. 

product  to  the  root  of  the  given  integer  number ;  the  sum  is  the 
root  required. 

Required  the  square  root  of  20*321. 

Square  root  of  21  =  4-5825 
Do.          20  =  44721 

•1104  x  -821  +  4-4721  =  4-5075384,  the 
square  root  required. 

Required  the  cube  root  of  16-42. 
Cube  root  of  17  =  2-5712 
Do.       16  =  2-5198 

•0514  x  -42  +  2-5198  =  2-541388,  the  cube 
root  required. 

To  find  the  squares  of  numbers  in  arithmetical  progression  ;  or, 
to  extend  the  foregoing  table  of  squares. 

RULE. — Find,  in  the  usual  way,  the  squares  of  the  first  two  num- 
bers, and  subtract  the  less  from  the  greater.  Set  down  the  square 
of  the  larger  number,  in  a  separate  column,  and  add  to  it  the  dif- 
ference already  found,  with  the  addition  of  2,  as  a  constant  quan- 
tity ;  the  product  will  be  the  square  of  the  next  following  number. 

The  square  of  1500 =  2250000 2250000 

The  square  of  1499 =  2247001 

Difference 2999  +  2=       3001 

The  square  of  1501 2253001 

Difference 3001  +  2  =        3003 

The  square  of  1502 2256004 

To  find  the  square  of  a  greater  number  than  is  contained  in  the  table. 

RULE  1. — If  the  number  required  to  be  squared  exceed  by  2,  3, 4, 
or  any  other  number  of  times,  any  number  contained  in  the  table, 
let  the  square  affixed  to  the  number  in  the  table  be  multiplied  by 
the  square  of  2,  3,  or  4,  &c.,  and  the  product  will  be  the  answer 
sought. 

Required  the  square  of  2595. 

2595  is  three  times  greater  than  865 ;  and  the  square  of  865, 
by  the  table,  is  748225. 

Then,  748225  x  32  =  6734025. 

RULE  2. — If  the  number  required  to  be  squared  be  an  odd  num- 
ber, and  do  not  exceed  twice  the  amount  of  any  number  contained 
in  the  table,  find  the  two  numbers  nearest  to  each  other,  which, 
added  together,  make  that  sum ;  then  the  sum  of  the  squares  of 
these  two  numbers,  by  the  table,  multiplied  by  2,  will  exceed  the 
square  required  by  1. 

Required  the  square  of  1865. 

The  two  nearest  numbers  (932  +  933)  =  1865. 

Then,  by  table  (9322  =  868624)  +  (9332  =  870489)  =  1739113  x 
2  =  3478226  -  1  =  3478225. 


RULES   FOR   SQUARES,   CUBES,    SQUARE   ROOTS,   ETC.          127 

To  find  the  cube  of  a  greater  number  than  is  contained  in  the  table. 
RULE. — Proceed,  as  in  squares,  to  find  how  many  times  the  num- 
ber required  to  be  cubed  exceeds  the  number  contained  in  the  table. 
Multiply  the  cube  of  that  number  by  the  cube  of  as  many  times  as 
the  number  sought  exceeds  the  number  in  the  table,  and  the  pro- 
duct will  be  the  answer  required. 

Required  the  cube  of  3984. 

3984  is  4  times  greater  than  996;  and  the  cube  of  996,  by  the 
table,  is  988047936. 

Then,  988047936  x  43  =  63235067904. 

To  find  the  square  or  cube  root  of  a  higher  number  than  is  in  the  table. 

RULE. — Refer  to  the  table,  and  seek  in  the  column  of  squares 
or  cubes  the  number  nearest  to  that  number  whose  root  is  sought, 
and  the  number  from  which  that  square  or  cube  is  derived  will  be 
the  answer  required,  when  decimals  are  not  of  importance. 

Required  the  square  root  of  542869. 

In  the  Table  of  Squares,  the  nearest  number  is  543169 ;  and 
the  number  from  which  that  square  has  been  obtained  is  737. 
Therefore,  v/542869  =  737  nearly. 

To  find  more  nearly  the  cube  root  of  a  higher  number  than  is  in 

the  table. 

RULE. — Ascertain,  by  the  table,  the  nearest  cube  number  to  the 
number  given,  and  call  it  the  assumed  cube. 

Multiply  the  assumed  cube,  and  the  given  number,  respectively, 
by  2  ;  to  the  product  of  the  assumed  cube  add  the  given  number, 
and  to  the  product  of  the  given  number  add  the  assumed  cube. 

Then,  by  proportion,  as  the  sum  of  the  assumed  cube  is  to  the 
sum  of  the  given  number,  so  is  the  root  of  the  assumed  cube  to 
the  root  of  the  given  number. 

Required  the  cube  root  of  412568555. 

By  the  table,  the  nearest  number  is  411830784,  and  its  cube 
root  is  744. 

Therefore,  411830784  x  2  +  412568555  =  1236230123. 
And,  412568555  x  2  +  411830784  =  1236967894. 
Hence,  as  1236230123  : 1236967894  : :  744  :  744-369,  very  nearly. 

To  find  the  square  or  cube  root  of  a  number  containing  decimals. 

RULE. — Subtract  the  square  root  or  cube  root  of  the  integer  of 
the  given  number  from  the  root  of  the  next  higher  number,  and 
multiply  the  difference  by  the  decimal  part.  The  product,  added  to 
the  root  of  the  integer  of  the  given  number  will  be  the  answer 
required. 

Required  the  square  root  of  321-62. 

x/321  =  17-9164729,  and  -v/322  =  17-9443584;  the  difference 
(-0278855)  x  -62  -f  17-9164729  =  17-9337619. 


128 


THE  PRACTICAL  MODEL  CALCULATOR. 


To  obtain  the  square  root  or  cube  root  of  a  number  containing  deci- 
mals, by  inspection. 

RULE. — The  square  or  cube  root  of  a  number  containing  deci- 
mals may  be  found  at  once  by  inspection  of  the  tables,  by  taking 
the  figures  cut  off  in  the  number,  by  the  decimal  point,  in  pairs 
if  for  the  square  root,  and  in  triads  if  for  the  cube  root.  The  fol- 
lowing example  will  show  the  results  obtained,  by  simple  inspec- 
tion of  the  tables,  from  the  figures  234,  and  from  the  numbers 
formed  by  the  addition  of  the  decimal  point  or  of  ciphers. 


Number. 

•00234 
•0234 
•2340 
2-34 
23-40 
234 
2340 
23400 


Square  Root. 

•0483735465* 
•152970585 
•483735465 
1-52970585 
4-83735465 
15-2970585 
48-3735465 
152-970585 


•132761439f 
•284| 
•61622401 
1-32761439 
2-860 
6-1622401 
13-2761439 
28-60 


To  find  the  cubes  of  numbers  in  arithmetical  progression,  or  to  extend 
the  preceding  table  of  cubes. 

RULE. — Find  the  cubes  of  the  first  two  numbers,  and  subtract 
the  less  from  the  greater.  Then,  multiply  the  least  of  the  two 
numbers  cubed  by  6,  add  the  product,  with  the  addition  of  6  as  a 
constant  quantity,  to  the  difference ;  and  thus,  adding  6  each  time 
to  the  sum  last  added,  form  a  first  series  of  differences. 

To  form  a  second  series  of  differences,  bring  down,  in  a  separate 
column,  the  cube  of  the  highest  of  the  above  numbers,  and  add 
the  difference  to  it.  The  amount  will  be  the  cube  of  the  next 
general  number. 

Required  the  cubes  of  1501,  1502,  and  1503. 


First  series  of  differences. 
By  Tab.  1500  ==  3375000000 
1499  =  3368254499 


1499x6+6  = 
9000  +  6  = 
9006  +  6  = 


6745501  difference. 
9000 


6754501  diff.  of 1500 
9006 


6763507  diff.  of 1501 
9012 


6772519  diff.  of 1502 


&c.,  &c. 


Second  series  of  differences. 

Then,  3375000000  Cube  of  1500 
Diff.  for  1500  =       6754501 

3381 754501  Cube  of  1501 
Diff.  for  1501  =       6763507 

3388518008  Cube  of  1502 
Diff.  for  1502=       6772519 


3395290527  Cube  of  1503 
&c.,  &c. 


*  Derived  from  -002340  by  means  of  2340. 

f  Derived  from  -002340  by  means  of  2340. 

j  The  nearest  result  by  simple  inspection  is  obtained  for  -023  by  23.  But  four 
places  correct  can  always  be  obtained  by  looking  in  the  table  of  cubes  for  the 
nearest  triad  or  triads,  in  this  instance  for  23400 ;  the  cube  beginning  with  the 
figures  23393  is  that  of  2860,  whence  -2860  is  true  to  the  last  place,  and  is  after- 
wards substituted. 


TABLE  OF  THE  FOURTH  AND  FIFTH  POWERS  OF  NUMBERS.     129 


TABLE  of  the  Fourth  and  Fifth  Powers  of  Numbers. 


Number. 

4th  Power. 

5th  Power. 

Number. 

4th  Power. 

5th  Power. 

1 

1 

1 

M 

33362176 

2535525376 

2 

16 

32 

77 

35153041 

2706784157 

3 

81 

243 

78 

37015056 

2887174368 

4 

256 

1024 

79 

38950081 

3077056399 

5 

625 

3125 

80 

40960000 

3276800000 

6 

1296 

7776 

81 

43046721 

3486784401 

7 

2401 

16807 

82 

45212176 

3707398432 

8 

4096 

32768 

83 

47458321 

3939040643 

9 

6561 

59049 

84 

49787136 

4182119424 

10 

10000 

100000 

85 

52200625 

4437053125 

11 

14641 

161051 

86 

54708016 

4704270176 

12 

20736 

248832 

87 

57289761 

4984209207 

13 

28561 

371293 

88 

59969536 

5277319168 

14 

38416 

537824 

89 

62742241 

5584059449 

15 

50625 

759375 

90 

65610000 

5904900000 

16 

65536 

1048576 

91 

68574961 

6240321451 

17 

83521 

1419857 

92 

71639296 

6590815232 

18 

104976 

1889568 

93 

74805201 

6596883693 

19 

130321 

2476099 

94 

78074896 

7339040224 

20 

160000 

3200000 

95 

81450625 

7737809375 

21 

194481 

4084101 

96 

84934656 

8153726976 

22 

234256 

5153632 

97 

88529281 

8587340257 

23 

279841 

6436343 

98 

92236816 

9039207968 

24 

331776 

7962624 

99 

96059601 

9509900499 

25 

390625 

9765625 

100 

100000000 

10000000000 

26 

456976 

11881376 

101 

104060401 

10510100501 

27 

531441 

14348907 

102 

108243216 

11040808032 

28 

614656 

17210368 

103 

112550881 

11592740743 

29 

707281 

20511149 

104 

116985856 

12166529024 

30 

810000 

24300000 

105 

121550625 

12762815625 

31 

923521 

28629151 

106 

126247696 

13382255776 

32 

1048576 

33554432 

107 

131079601 

14025517307 

33 

1185921 

39135393 

108 

136048896 

14693280768 

34 

1336336 

45435424 

109 

141158161 

15386239549 

35 

1500625 

52521875 

110 

146410000 

16105100000 

36 

1679616 

60466176 

111 

151807041 

16850581551 

37 

1874161 

69343957 

112 

157351936 

17623416832 

38 

2085136 

79235168 

113 

163047361 

18424351793 

39 

2313441 

90224199 

114 

168896016 

19254145824 

40 

2560000 

102400000 

115 

174900625 

20113571875 

41 

2825761 

115856201 

116 

181063936 

21003416576 

42 

3111696 

130691232 

117 

187388721 

21924480357 

43 

3418801 

147008443 

118 

193877776 

22877577568 

44 

3748096 

164916224 

119 

200533921 

23863536599 

45 

4100625 

184528125 

120 

207360000 

24883200000 

46 

4477456 

205962976 

121 

214358881 

25937424601 

47 

4879681 

229345007 

122 

221533456 

27027081632 

48 

5308416 

254803968 

123 

228886641 

28153056843 

49 

5764801 

282475249 

124 

236421376 

29316250624 

50 

6250000 

312500000 

125 

244140625 

30517578125 

51 

6765201 

345025251 

126 

252047376 

31757969376 

52 

7311616 

380204032 

127 

260144641 

33038369407 

53 

7890481 

418195493 

128 

268435456 

34359738368 

54 

8503056 

459165024 

129 

276922881 

35723051649 

55 

9150625 

503284375 

130 

285610000 

37129300000 

56 

9834496 

550731776 

131 

294499921 

38579489651 

57 

10556001 

601692057 

132 

303595776 

40074642432 

58 

11316496 

656356768 

133 

312900721 

41615795893 

59 

12117361 

714924299 

134 

322417936 

43204003424 

60 

12960000 

777600000 

135 

332150625 

44840334375 

61 

13845841 

844596301 

136 

342102016 

46525874176 

62 

14776336 

916132832 

137 

352275361 

48261724457 

63 

15752961 

992436543 

138 

362673936 

50049003168 

64 

16777216 

1073741824 

139 

373301041 

51888844699 

65 

17850625 

1160290625 

140 

384160000 

53782400000 

66 

18974736 

1252332576 

141 

395254161 

55730836701 

67 

20151121 

1350125107 

142 

406586896 

57735339232 

68 

21381376 

1453933568 

143 

418161601 

59797108943 

69 

22667121 

1564031349 

144 

429981696 

61917364224 

70 

24010000 

1680700000 

145 

442050625 

64097340625 

71 

25411681 

1804229351 

146 

454371856 

66338290976 

72 

26873856 

1934917632 

147 

466948881 

68641485507 

73 

28398241 

2073071593 

148 

479785216 

71008211968 

74 

29986576 

2219006624 

149 

492884401 

73439775749 

75 

31640625 

2373046875 

150 

506250000 

76937500000 

130 


THE   PRACTICAL   MODEL   CALCULATOR. 


TABLE  of  Hyperbolic  Logarithms. 


f  —  — 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

ToT 

•0099503 

1-58 

•4574248 

2-15 

•7654678 

2-72 

1-0006318 

1-02 

•0198026 

1-59 

•4637340 

2-16 

•7701082 

2-73 

1-0043015 

1-03 

•0295588 

1-60 

•4700036 

2-17 

•7747271 

2-74 

1-0079579 

1-04 

•0392207 

1-61 

•4762341 

2-18 

•7793248 

2-75 

1-0116008 

1-05 

•0487902 

1-62 

•4824261 

2-19 

•7839015 

2-76 

1-0152306 

1-06 

•0582689 

1-63 

•4885800 

2-20 

•7884573 

2-77 

1-0188473 

1-07 

•0676586 

1-64 

•4946962 

2-21 

•7929925 

2-78 

1  -0224509 

1-08 

•0769610 

1-65 

•5007752 

2-22 

•7975071 

2-79 

1-0260415 

1-09 

-0861777 

1-66 

•5068175 

2-23 

•8020015 

2-80 

1-0296194 

1-10 

-0953102 

1-67 

•5128236 

2-24 

•8064758 

2-81 

1-0331844 

1-11 

•1043600 

1-68 

•5187937 

2-25 

•8109302 

2-82 

1-0367368 

1-12 

•1133287 

1-69 

•5247285 

2-26 

•8153648 

2-83 

1-0402766 

1-13 

•1222176 

•70 

•5306282 

2-27 

•8197798 

2-84 

1-0438040 

1-14 

•1310283 

•71 

•5364933 

2-28 

•8241754 

2-85 

1-0473189 

1-16 

•1397619 

•72 

•5423242 

2-29 

•8285518 

2-86 

1-0508216 

1-16 

•1484200 

•73 

•5481214 

2-30 

•8329091 

2-87 

1-0543120 

1-17 

•1570037 

•74 

•5538851 

2-31 

•8372475 

2-88 

1-0577902 

1-18 

•1655144 

•75 

•5596157 

2-32 

•8415671 

2-89 

1-0612564 

1-19 

•1739533 

•76 

•5653138 

2-33 

•8458682 

2-90 

1-0647107 

1-20 

•1823215 

•77 

•5709795 

2-34 

•8501509 

2-91 

1-0681530 

1-21 

•1906203 

•78 

•5766133 

2-35 

•8544153 

2-92 

1-0715836 

1-22 

•1988508 

•79 

•5822156 

2-36 

•8586616 

2-93 

1-0750024 

1-23 

•2070141 

•80 

•5877866 

2-37 

•8628899 

2-94 

1-0784095 

1-24 

•2151113 

•81 

•5933268 

2-38 

•8671004 

2-95 

•0818051 

1-25 

•2231435 

•82 

•5988365 

2-39 

•8712933 

2-96 

•0851892 

1-26 

•2311117 

•83 

•6043159 

2-40 

•8754687 

2-97 

•0885619 

1-27 

•2390169 

•84 

•6097655 

2-41 

•8796267 

2-98 

•0919233 

1-28 

•2468600 

•85 

•6151856 

2-42 

•8837675 

2-99 

•0952733 

1-29 

•2546422 

•86 

•6205764 

2-43 

•8878912 

3-00 

•0986123 

1-30 

•2623642 

•87 

•6259384 

2-44 

•8919980 

3-01 

•1019400 

1-31 

•2700271 

•88 

•6312717 

2-45 

•8960880 

3-02 

•1052568 

1-32 

•2776317 

•89 

•6365768 

2-46 

•9001613 

3-03 

•1085626 

1-33 

•2851789 

•90 

•6418538 

2-47 

•9042181 

3-04 

•1118575 

1-34 

•2926696 

•91 

•6471032 

2-48 

•9082585 

3-05 

•1151415 

1-35 

•3001045 

•92 

•6523251 

2-49 

•9122826 

3-06 

•1184149 

1-36 

•3074846 

•93 

•6575200 

2-50 

•9162907 

3-07 

•1216775 

1-37 

•3148107 

1-94 

•6626879 

2-51 

•9202827 

3-08 

•1249295 

1-38 

•3220834 

1-95 

•6678293 

2-52 

•9242589 

3-09 

•1281710 

1-39 

•3293037 

1-96 

•6729444 

2-53 

•9282193 

3-10 

•1314021 

1-40 

•3364722 

1-97 

•6780335 

2-54 

•9321640 

3-11 

1-1346227 

1-41 

•3435897 

1-98 

•6830968 

2-55 

•9360933 

8-12 

1-1378330 

1-42 

•3506568 

1-99 

•6881346 

2-56 

•9400072 

3-13 

1-1410330 

1-43 

•3576744 

2-00 

•6931472 

2-57 

•9439058 

3-14 

1-1442227 

1-44 

•3646431 

2-01 

•6981347 

2-58 

•9477893 

3-15 

1-1474024 

1-45 

•3715635 

2-02 

•7030974 

2-59 

•9516578 

3-16 

1-1505720 

1-46 

-3784364 

2-03 

•7080357 

2-60 

•9555114 

3-17 

1-1537315 

1-47 

•3852624 

2-04 

•7129497 

2-61 

•9593502 

3-18 

1-1568811 

1-48 

•3920420 

2-05 

•7178397 

2-62 

•9631743 

3-19 

1-1600209 

1-49 

•3987761 

2-06 

•7227059 

2-63 

•9669838 

3-20 

1-1631508 

1-50 

•4054651 

2-07 

•7275485 

2-64 

•9707789 

3-21 

1-1662709 

1-61 

•4121096 

2-08 

•7323678 

2-65 

•9745596 

3-22 

1-1693813 

1-52 

•4187103 

2-09 

•7371640 

2-66 

•9783261 

3-23 

1-1724821 

1-53 

•4252677 

2-10 

•7419373 

2-67 

•9820784 

3-24 

1-1755733 

1-54 

•4317824 

2-11 

•7466879 

2-68 

•9858167 

3-25 

1-1786549 

1-55 

•4382549 

2-12 

•7514160 

2-69 

•9895411 

3-26 

1-1817271 

1-56 

•4446858 

2-13 

•7561219 

2-70 

•9932517 

3-27 

1-1847899 

1-57 

•4510756 

2-14 

•7608058 

2-71 

•9969486 

3-28 

1-1878434 

TABLE   OF  HYPERBOLIC   LOGARITHMS. 


131 


N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

3-29 

1-1908875 

3-91 

1-3635373 

4-53 

1-5107219 

5-15 

1-6389967 

3-30 

1-1939224 

3-92 

1-3660916 

4-54 

1-5129269 

5-16 

1-6409365 

3-31 

1-1969481 

3-93 

1-3686394 

4-55 

1-5151272 

5-17 

1-6428726 

3-32 

1-1999647 

3-94 

1-3711807 

4-56 

1-5173226 

5-18 

1-6448050 

3-33 

1-2029722 

3-95 

1-3737156 

4-57 

1-5195132 

5-19 

1-6467336 

3-34 

1-2059707 

3-96 

1-3762440 

4-58 

1-5216990 

5-20 

1-6486586 

3-35 

1-2089603 

3-97 

1-3787661 

4-59 

1-5238800 

5-21 

1-6505798 

3-36 

1-2119409 

3-98 

•3812818 

4-60 

1-5260563 

5-22 

1-6524974 

3-37 

1-2149127 

3-99 

•3837912 

4-61 

1-5282278 

5-23 

1-6544112 

3-38 

1-2178757 

4-00 

•3862943 

4-62 

1-5303947 

5-24 

1-6563214 

3-39 

1-2208299 

4-01 

•3887912 

4-63 

1-5325568 

5-25 

1-6582280 

3-40 

1-2237754 

4-02 

•3912818 

4-64 

1-5347143 

5-26 

1-6601310 

3-41 

1-2267122 

4-03 

•3937663 

4-65 

1-5368672 

5-27 

1-6620303 

3-42 

1-2296405 

4-04 

1-3962446 

4-66 

1-5390154 

5-28 

1-6639260 

3-43 

1-2325605 

4-05 

1-3987168 

4-67 

1-5411590 

5-29 

1-6658182 

3-44 

1-2354714 

4-06 

1-4011829 

4-68 

1-5432981 

5-30 

1-6677068 

3-45 

1-2383742 

4-07 

1-4036429 

4-69 

1-5454325 

5-31 

1-6695918 

3-46 

1-2412685 

4-08 

1-4060969 

4-70 

1-5475625 

5-32 

1-6714733 

347 

1-2441545 

4-09 

1-4085449 

4-71 

1-5496879 

5-33 

1-6733512 

3-48 

1-2470322 

4-10 

1-4109869 

4-72 

1-5518087 

5-34 

1-6752256 

3-49 

1-2499017 

4-11 

1-4134230 

4-73 

1-5539252 

5-35 

1-6770965 

3-50 

1-2527629 

4-12 

1-4168531 

4-74 

1-5560371 

5-36 

1-6789639 

3-51 

1-2556160 

4-13 

1-4182774 

4-75 

1-5581446 

5-37 

1-6808278 

3-52 

1-2584609 

4-14 

1-4206957 

4-76 

1-5602476 

5-38 

1-6826882 

3-53 

1-2612978 

4-15 

1-4231083 

4-77 

1-5623462 

5-39 

1-6845453 

3-54 

1-2641266 

4-16 

1-4255150 

4-78 

1-5644405 

5-40 

1-6863989 

3-55 

1-2669475 

4-17 

1-4279160 

4-79 

1-5665304 

5-41 

1-6882491 

3-56 

1-2697605 

4-18 

1-4303112 

4-80 

1-5686159 

5-42 

1-6900958 

3-57 

1-2725655 

4-19 

1-4327007 

4-81 

1-5706971 

5-43 

1-6919391 

3-58 

1-2753627 

4-20 

1-4350845 

4-82 

1-5727739 

5-44 

1-6937790 

3-59 

1-2781521 

4-21 

1-4374626 

4-83 

1-5748464 

5-45 

1-6956155 

3-60 

1-2809338 

4-22 

1-4398351 

4-84 

1-5769147 

5-46 

1-6974487 

3-61 

1-2837077 

4-23 

1-4422020 

4-85 

1-5789787 

5-47 

1-6992786 

3-62 

1-2864740 

4-24 

1-4445632 

4-86 

1-5810384 

5-48 

1-7011051 

3-63 

1-2892326 

4-25 

1-4469189 

4-87 

1-5830939 

5-49 

1-7029282 

3-64 

1-2919836 

4-26 

1-4492691 

4-88 

1-5851452 

5-50 

1-7047481 

3-65 

1-2947271 

4-27 

1-4516138 

4-89 

1-5871923 

5-51 

1-7065646 

3-66 

1-2974631 

4-28 

1-4539530 

4-90 

1-5892352 

5-52 

1-7083778 

3-67 

1-3001916 

4-29 

1-4562867 

4-91 

1-5912739 

5-53 

1-7101878 

3-68 

1-3029127 

4-30 

1  -4586149 

4-92 

1-5933085 

5-54 

1.7119944 

3-69 

1-3056264 

4-31 

1-4609379 

4-93 

1-5953389 

5-55 

1-7137979 

3-70 

1-3083328 

4-32 

1-4632553 

4-94 

1-5973653 

5-56 

1-7155981 

3-71 

1-3110318 

4-33 

1-4655675 

4-95 

1-5993875 

5-57 

1-7173950 

3-72 

1-3137236 

4-34 

1-4678743 

4-96 

1-6014057 

5-58 

1-7191887 

3-73 

1-3164082 

4-35 

1-4701758 

4-97 

1-6034198 

5-59 

1-7209792 

3-74 

1-3190856 

4-36 

1-4724720 

4-98 

1-6054298 

5-60 

1-7227666 

3-75 

1-3217558 

4-37 

1-4747630 

4-99 

1-6074358 

5-61 

1-7245507 

3-76 

1-3244189 

4-38 

1-4770487 

5-00 

1-6094379 

5-62 

1-7263316 

3-77 

1-3270749 

4-39 

1-4793292 

5-01 

1-6114359 

5-63 

1-7281094 

3-78 

1-3297240 

4-40 

1-4816045 

5-02 

1-6134300 

5-64 

1-7298840 

3-79 

1-3323660 

4-41 

1-4838746 

5-03 

1-6154200 

5-65 

1-7316555 

3-80 

1-3350010 

4-42 

1-4861396 

5-04 

1-6174060 

5-66 

1-7334238 

3-81 

1-3376291 

4-43 

1-4883995 

5-05 

1-6193882 

5-67 

1-7351891 

3-82 

1-3402504 

4-44 

1-4906543 

5-06 

1-6213664 

5-68 

1-7369512 

3-83 

1-3428648 

4-45 

1-4929040 

5-07 

1-6233408 

5-69 

1-7387102 

3-84 

1-3454723 

4-46 

1-4951487 

5-08 

1-6253112 

5-70 

1-7404661 

3-85 

1-3480731 

4-47 

1-4973883 

5-09 

1-6272778 

5-71 

1-7422189 

3-86 

1-3506671 

4-48 

1-4996230 

5-10 

1-6292405 

5-72 

1-7439687 

3-87 

1-3532544 

4-49 

1-5018527 

5-11 

1-6311994 

5-73 

1-7457155 

3-88 

1-3558351 

4-50 

1-5040774 

5-12 

1-6331544 

5-74 

1-7474591 

3-89 

1-3584091 

4-51 

1-5062971 

5-13 

1-6351056 

5-75 

1-7491998 

3-90 

1-3609765 

4-52 

1-5085119 

5-14 

1-6370530 

5-76 

1-7509374 

132 


THE   PRACTICAL   MODEL   CALCULATOR, 


N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm* 

5-77 

1-7526720 

6-39 

1-8547342 

7-01 

1-9473376 

7-63 

2-0320878 

5-78 

1-7544036 

6-40 

1-8562979 

7-02 

1-9487632 

7-64 

2-0333976 

5-79 

1-7561323 

6-41 

1-8578592 

7-03 

1-9501866 

7-65 

2-0347056 

5-80 

1-7578579 

6-42 

1-8594181 

7-04 

1-9516080 

7-66 

2-0360119 

5-81 

1-7595805 

6-43 

1-8609745 

7-05 

1-9530275 

7-67 

2-0373166 

5-82 

1-7613002 

6-44 

1-8625285 

7-06 

1-9544449 

7-68 

2-0386195 

5-83 

1-7630170 

6-45 

1-8640801 

7-07 

1-9558604 

7-69 

2-0399207 

5-84 

1-7647308 

6-46 

1-8656293 

7-08 

•9572739 

7-70 

2-0412203 

5-85 

1-7664416 

6-47 

1-8671761 

7-09 

•9586853 

7-71 

2-0425181 

5-86 

1-7681496 

6-48 

1-8687205 

7-10 

•9600947 

7-72 

2-0438143 

5-87 

1-7698546 

6-49 

1-8702625 

7-11 

•9615022 

7-73 

2-0451088 

5-88 

1-7715567 

6-50 

1-8718021 

7-12 

•9629077 

7-74 

2-0464016 

5-89 

1-7732559 

6-51 

1-8733394 

7-13 

•9643112 

7-75 

2-0476928 

5-90 

1-7749523 

6-52 

1-8748743 

7-14 

•9657127 

7-76 

2-0489823 

5-91 

1-7766458 

6-53 

1-8764069 

7-15 

•9671123 

7-77 

2-0502701 

5-92 

1-7783364 

6-54 

1-8779371 

7-16 

•9685099 

7-78 

2-0515563 

5-93 

1-7800242 

6-55 

1-8794650 

7-17 

•9699056 

7-79 

2-0528408 

5-94 

1-7817091 

6-56 

1-8809906 

7-18 

•9712993 

7-80 

2-0541237 

5-95 

1-7833912 

6-57 

1-8825138 

7-19 

•9726911 

7-81 

2-0554049 

5-96 

1-7850704 

6-58 

1-8840347 

7-20 

•9740810 

7-82 

2-0566845 

5-97 

1-7867469 

6-59 

1-8855533 

7-21 

•9754689 

7-83 

2-0579624 

5-98 

1-7884205 

6-60 

1-8870696 

7-22 

•9768549 

7-84 

2-0592388 

5-99 

1-7900914 

6-61 

1-8885837 

7-23 

•9782390 

7-85 

2-0605135 

6-00 

1-7917594 

6-62 

1-8900954 

7-24 

•9796212 

7-86 

2-0617866 

6-01 

1-7934247 

6-63 

1-8916048 

7-25 

1-9810014 

7-87 

2-0630580 

6-02 

1-7950872 

6-64 

1-8931119 

7-26 

1-9823798 

7-88 

2-0643278 

6-03 

1-7967470 

6-65 

1-8946168 

7-27 

1-9837562 

7-89 

2-0655961 

6-04 

1-7984040 

6-66 

1-8961194 

7-28 

1-9851308 

7-90 

2-0668627 

6-05 

1-8000582 

6-67 

1-8976198 

7-29 

1-9865035 

7-91 

2-0681277 

6-06 

1-8017098 

6-68 

1-8991179 

7-30 

1-9878743 

7-92 

.  2-0693911 

6-07 

1-8033586 

6-69 

1-9006138 

7-31 

1-9892432 

7-93 

2-0706530 

6-08 

1-8050047 

6-70 

1-9021075 

7-32 

1-9906103 

7-94 

2-0719132 

6-09 

1-8066481 

6-71 

1-9035989 

7-33 

1-9919754 

7-95 

2-0731719 

6-10 

1-8082887 

6-72 

1-9050881 

7-34 

1-9933387 

7-96 

2-0744290 

6-11 

1-8099267 

6-73 

1-9065751 

7-35 

1-9947002 

7-97 

2-0756845 

6-12 

1-8115621 

6-74 

1-9080600 

7-36 

1-9960599 

7-98 

2-0769384 

6-13 

1-8131947 

6-75 

1-9095425 

7-37 

1-9974177 

7-99 

2-0781907 

6-14 

1-8148247 

6-76 

1-9110228 

7-38 

1-9987736 

8-00 

2-0794415 

6-15 

1-8164520 

6-77 

1-9125011 

7-39 

2-0001278 

8-01 

2-0806907 

6-16 

1-8180767 

6-78 

1-9139771 

7-40 

2-0014800 

8-02 

2-0819384 

6-17 

1-8196988 

6-79 

1-9154509 

7-41 

2-0028305 

8-03 

2-0831845 

6-18 

1-8213182 

6-80 

1-9169226 

7-42 

2-0041790 

8-04 

2-0844290 

6-19 

1-8229351 

6-81 

1-9183921 

7-43 

2-0055258 

8-05 

2-0856720 

6-20 

1-8245493 

6-82 

1-9198594 

7-44 

2-0068708 

8-06 

2-0869135 

6-21 

1-8261608 

6-83 

1-9213247 

7-45 

2-0082140 

8-07 

2-0881534 

6-22 

1-8277699 

6-84 

1-9227877 

7-46 

2-0095553 

8-08 

2-0893918 

6-23 

1-8293763 

6-85 

1-9242486 

7-47 

2-0108949 

8-09 

2-0906287 

6-24 

1-8309801 

6-86 

1-9257074 

7-48 

2-0122327 

8-10 

2-0918640 

6-25 

1-8325814 

6-87 

1-9271641 

7-49 

2-0135687 

8-11 

2-0930984 

6-26 

1-8341801 

6-88 

1-9286186 

7-50 

2-0149030 

8-12 

2-0943306 

6-27 

1-8357763 

6-89 

1-9300710 

7-51 

2-0162354 

8-13 

2-0955613 

6-28 

1-8373699 

6-90 

1-9315214 

7-52 

2-0175661 

8-14 

2-0967905 

6-29 

1-8389610 

6-91 

1-9329696 

7-53 

2-0188950 

8-15 

2-0980182 

6-30 

1-8405496 

6-92 

1-9344157 

7-64 

2-0202221 

8-16 

2-0992444 

6-31 

1-8421356 

6-93 

1-9358598 

7  -55 

2-0215475 

8-17 

2-1004691 

6-32 

1-8437191 

6-94 

1-9373017 

7-56 

2-0228711 

8-18 

2-1016923 

G-33 

1-8453002 

6-95 

1-9387416 

7-57 

2-0241929 

8-19 

2-1029140 

6-34 

1-8468787 

6-96 

1-9401794 

7-58 

2-0255131 

8-20 

2-1041341 

6-35 

1-8484547 

6-97 

1-9416152 

7-59 

2-0268315 

8-21 

2-1053529 

6-36 

1-8500283 

6-98 

1-9430489 

7-60 

2-0281482 

8-22 

2-1065702 

6-37 

1-8515994 

6-99 

1-9444805 

7-61 

2-0294631 

8-23 

2-1077861 

6-38 

1-8531680 

7-00 

1-9459101 

7-62 

2-0307763 

8-24 

2-1089998 

TABLE   OF   HYPERBOLIC   LOGARITHMS. 


133 


N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

8-25 

2-1102128 

8-69 

2-1621729 

9-13 

2-2115656 

9-57 

2-2586332 

8-26 

2-1114243 

8-70 

2-1633230 

9-14 

2-2126603 

9-58 

2-2596776 

8-27 

2-1126343 

8-71 

2-1644718 

9-15 

2-2137538 

9-59 

2-2607209 

8-28 

2-1138428 

8-72 

2-1656192 

9-16 

2-2148461 

9-60 

2-2617631 

8-29 

2-1150499 

8-73 

2-1667653 

9-17 

2-2159372 

9-61 

2-2628042 

8-30 

2-1162555 

8-74 

2-1679101 

9-18 

2-2170272 

9-62 

2-2638442 

8-31 

2-1174596 

8-75 

2-1690536 

9-19 

2-2181160 

9-63 

2-2648832 

8-32 

2-1186622 

8-76 

2-1701959 

9-20 

2-2192034 

9-64 

2-2659211 

8-33 

2-1198634 

8-77 

2-1713367 

9-21 

2-2202898 

9-65 

2-2669579 

8-34 

2-1210632 

8-78 

2-1724763 

9-22 

2-2213750 

9-66 

2-2679936 

8-35 

2-1222615 

8-79 

2-1736146 

9-23 

2-2224590 

9-67 

2-2690282 

8-36 

2-1234584 

8-80 

2-1747517 

9-24 

2-2235418 

9-68 

2-2700618 

8-37 

2-1246539 

8-81 

2-1758874 

9-25 

2-2246235 

9-69 

2-2710944 

8-38 

2-1258479 

8-82 

2-1770218 

9-26 

2-2257040 

9-70 

2-2721258 

8-39 

2-1270405 

8-83 

2-1781550 

9-27 

2-2267833 

9-71 

2-2731562 

8-40 

2-1282317 

8-84 

2-1792868 

9-28 

2-2278615 

9-72 

2-2741856 

8-41 

2-1294214 

8-85 

2-1804174 

9-29 

2-2289385 

9-73 

2-2752138 

8-42 

2-1306098 

8-86 

2-1815467 

9-30 

2-2300144 

9-74 

2-2762411 

8-43 

2-1317967 

8-87 

2-1826747 

9-31 

2-2310890 

9-75 

2-2772673 

8-44 

2-1329822 

8-88 

2-1838015 

9-32 

2-2321626 

9-76 

2-2782924 

8-45 

2-1341664 

8-89 

2-1849270 

9-33 

2-2332350 

9-77 

2-2793165 

8-46 

2-1353491 

8-90 

2-1860512 

9-34 

2-2343062 

9-78 

2-2803395 

8-47 

2-1365304 

8-91 

2-1871742 

9-35 

2-2353763 

9-79 

2-2813614 

8-48 

2-1377104 

8-92 

2-1882959 

9-36 

2-2364452 

9-80 

2-2823823 

8-49 

2-1388889 

8-93 

2-1894163 

9-37 

2-2375130 

9-81 

2-2834022 

8-50 

2-1400661 

8-94 

2-1905355 

9-38 

2-2385797 

9-82 

2-2844211 

8-51 

2-1412419 

8-95 

2-1916535 

9-39 

2-2396452 

9-83 

2-2854389 

8-52 

2-1424163 

8-96 

2-1927702 

9-40 

2-2407096 

9-84 

2-2864556 

8-53 

2-1435893 

8-97 

2-1938856 

9-41 

2-2417729 

9-85 

2-2874714 

8-54 

2-1447609 

8-98 

2-1949998 

9-42 

2-2428350 

9-86 

2-2884861 

8-55 

2-1459312 

8-99 

2-1961128 

9-43 

2-2438960 

9-87 

2-2894998 

8-56 

2-1471001 

9-00 

2-1972245 

9-44 

2-2449559 

9-88 

2-2905124 

8-57 

2-1482676 

9-01 

2-1983350 

9-45 

2-2460147 

9-89 

2-2915241 

8-58 

2-1494339 

9-02 

2-1994443 

9-46 

2-2470723 

9-90 

2-2925347 

8-59 

2-1505987 

9-03 

2-2005523 

9-47 

2-2481288 

9-91 

2-2635443 

8-60 

2-1517622 

9-04 

2-2016591 

9-48 

2-2491843 

9-92 

2-2945529 

8-61 

2-1529243 

9-05 

2-2027647 

9-49 

2-2502386 

9-93 

2-2955604 

8-62 

2-1540851 

9-06 

2-2038691 

9-50 

2-2512917 

9-94 

2-2965670 

8-63 

2-1552445 

9-07 

2-2049722 

9-51 

2-2523438 

9-95 

2-2975725 

8-64 

2-1564026 

9-08 

2-2060741 

9-52 

2-2533948 

9-96 

2-2985770 

8-65 

2-1575593 

9-09 

2-2071748 

9-53 

2-2544446 

9-97 

2-2995806 

8-66 

2-1587147 

9-10 

2-2082744 

9-54 

2-2554934 

9-98 

2-3005831 

8-67 

2-1598687 

9-11 

2-2093727 

9-55 

2-2565411 

9-99 

2-3015846 

8-68 

2-1610215 

9-12 

2-2104697 

9-56 

2-2575877 

10-00 

2-3025851 

Logarithms  were  invented  by  Juste  Byrge,  a  Frenchman,  and 
not  by  Napier.  See  "Biographic  Universelle,"  "  The  Calculus  of 
Form,"  article  822,  and  "The  Practical,  Short,  and  Direct  Method 
of  Calculating  the  Logarithm  of  any  given  Number  and  the  Number 
corresponding  to  any  given  Logarithm,"  discovered  by  Oliver  Byrne, 
the  author  of  the  present  work.  Juste  Byrge  also  invented  the 
proportional  compasses,  and  was  a  profound  astronomer  and  ma- 
thematician. The  common  Logarithm  of  a  number  multiplied  by 
2-302585052994  gives  the  hyperbolic  Logarithm  of  that  number. 
The  common  Logarithm  of  2-22  is  -346353  .-.  2-302585  X  -346353 
=  -Y975071  the  hyperbolic  Logarithm.  The  application  of  Loga- 
rithms to  the  calculations  of  the  Engineer  will  be  treated  of  here- 
after. 

M 


134 


THE   PRACTICAL  MODEL   CALCULATOR. 


COMBINATIONS   OF   ALGEBRAIC    QUANTITIES. 

THE  following  practical  examples  will  serve  to  illustrate  the 
method  of  combining  or  representing  numbers  or  quantities  alge- 
braically ;  the  chief  object  of  which  is,  to  help  the  memory  with 
respect  to  the  use  of  the  signs  and  letters,  or  symbols. 

Let  a  =  6,  £  =  4,  c  =  3,  d  =  2,  e  =  1,  and/  =  0. 
Then  will,     (1)  2a  -f  b  =  12  +  4  =  16. 

(2)  ab  +  2c  -  d  =  24  +  6  -  2  =  28. 

(8)  a2-b2  +  e  +/=  36  -  16  +  1-f-  0  =  21. 

(4)  b*  x  (a  -  b)  =  16  x  (6  -  4)  =  16  x  2  =  32. 

(5)  Babe  -  7de  =  216  -  14  =  202. 

(6)  2  (a  -  b)  (5c  -  2<2)  =  (12  -  8)  x  (15  -  4)  =  44. 


(a  -  e)  = 

(8)  v/  (a2  -  262)  -f  d  -/ 

(9)  3a6  -  (a  -  b  -  c  +  c?) 
(10)  Bab  -  (a  -  b  -  c  -  d) 

(11) 


x  (6  -  3)  =  4  x  3  =  12. 

(36  -  32)  -f  2  -  0  =  4. 
72  -  1  =  71. 

72  -f  3  =  75. 

x  (3  +  2)  =  15. 


In  solving  the  following  questions,  the  letters  a,  5,  c,  &c.  are 
supposed  to  have  the  same  values  as  before,  namely,  6,  4,  3,  &c.  ; 
but  any  other  values  might  have  been  assigned  to  them  ;  therefore, 
do  not  suppose  that  a  must  necessarily  be  6,  nor  that  b  must  be  4, 
for  the  letter  a  may  be  put  for  any  known  quantity,  number,  or 
magnitude  whatever  ;  thus  a  may  represent  10  miles,  or  50  pounds^ 
or  any  number  or  quantity,  or  it  may  represent  1  globe,  or  2  cubic 
feet,  &c.  ;  the  same  may  be  said  of  b,  or  any  other  letter. 


(1)  a  +  b  -  c  = 

(2)  Zbc  -  d  +  e 

(8)  2a2  +  <,  _  d 

(4)  £  x  (6  _  , 
6 

(5)  ScftZ  -  a2 


7. 


(6)  4  (a2  -  62)  (*-«)* 


35. 

=  79. 


x 


=  27.      W 
62.      (1°) 


-  (^2-^-e)  =570. 
>/4a2 


In  the  use  of  algebraic  symbols,  3 
thing  as  3  (4a  —  b)^. 

4  (c  +  d)*  (a  +  b)^,  or  4  X  ^T 
same  thing  as4->/c  +  ^  •  -^  a  +  6. 


Xd~  °2  = 
4a  —  6  signifies  the  same 

X  a  +  $,  signifies  the 


135 


THE   STEAM  ENGINE. 


THE  particular  example  which  we  shall  select  is  that  of  an 
engine  having  8  feet  stroke  and  64  inch  cylinder. 

The  breadth  of  the  web  of  the  crank  at  the  paddle  centre  is  the 
breadth  which  the  web  would  have  if  it  were  continued  to  the  paddle 
centre.  Suppose  that  we  wished  to  know  the  breadth  of  the  web  of 
crank  of  an  engine  whose  stroke  is  8  feet  and  diameter  of  cylinder  64 
inches.  The  proper  breadth  of  the  web  of  crank  at  paddle  centre 
would  in  this  case  be  about  18  inches. 

To  find  the  breadth  of  crank  at  paddle  centre. — Multiply  the 
square  of  the  length  of  the  crank  in  inches  by  1*561,  and  then 
multiply  the  square  of  the  diameter  of  cylinder  in  inches  by  -1235  ; 
multiply  the  square  root  of  the  sum  of  these  products  by  the  square 
of  the  diameter  of  the  cylinder  in  inches ;  divide  the  product  by 
45 ;  finally  extract  the  cube  root  of  the  quotient.  The  result  is 
the  breadth  of  the  web  of  crank  at  paddle  centre. 

Thus,  to  apply  this  rule  to  the  particular  example  which  we  have 
selected,  we  have 

48  =  length  of  crank  in  inches. 
48 

2304 

1-561  =  constant  multiplier. 

3596-5 
505-8  found  below. 


4102-3 


64  =  diameter  of  cylinder. 
64 

4096 
•1235  =  constant  multiplier. 

505-8 


and  ^4102-3  =  64-05  nearly. 

4096  =  square  of  the  diameter  of  the  cylinder. 

45)  262348-5 

5829-97     

and  ^5829-97  =  18  nearly. 

Suppose  that  we  wished  the  proper  thickness  of  the  large  eye  of 
crank  for  an  engine  whose  stroke  is  8  feet  and  diameter  of  cylinder 
64  inches.  The  proper  thickness  for  the  large  eye  of  crank  is 
5-77  inches. 


136          THE  PRACTICAL  MODEL  CALCULATOR. 

KULE. —  To  find  the  thickness  of  large  eye  of  crank. — Multiply  the 
square  of  the  length  of  the  crank  in  inches  by  1-561,  and  then  mul- 
tiply the  square  of  the  diameter  of  the  cylinder  in  inches  by  -1235 ; 
multiply  the  sum  of  these  products  by  the  square  of  the  diameter  of 
the  cylinder  in  inches  ;  afterwards,  divide  the  product  by  1828-28  ; 
divide  this  quotient  by  the  length  of  the  crank  in  inches ;  finally 
extract  the  cube  root  of  the  quotient.  The  result  is  the  proper 
thickness  of  the  large  eye  of  crank  in  inches. 

Thus,  to  apply  this  rule  to  the  particular  example  which  we  have 
selected,  we  have 

48  =  length  of  crank  in  inches. 

48 


2304 
1-561  constant  multiplier. 

3596-5 

505-8 

4102-3 

64  =  diameter  of  cylinder  in  inches. 
64 


4096 

•1235  =  constant  multiplier. 

50^8 

4102-3 
4096  =  square  of  diameter. 

48)  16803020-8 

1828-28)  350062-94 

191-47 


and  ^191-47  =  5-77  nearly. 

The  proper  thickness  of  the  web  of  crank  at  paddle  shaft  centre 
is  the  thickness  which  the  web  ought  to  have  if  continued  to  centre 
of  the  shaft.  Suppose  that  it  were  required  to  find  the  proper 
thickness  of  web  of  crank  at  shaft  centre  for  an  engine  whose 
stroke  is  8  feet  and  diameter  of  cylinder  64  inches.  The  proper 
thickness  of  the  web  at  shaft  centre  in  this  case  would  be  8-97 
inches. 

RULE. —  To  find  the  thickness  of  the  web  of  crank  at  paddle  shaft 
centre. — Multiply  the  square  of  the  length  of  crank  in  inches  by 
1-561,  and  then  multiply  the  square  of  the  diameter  in  inches  by 
•1235 ;  multiply  the  square  root  of  the  sum  of  these  products  by  the 
square  of  the  diameter  of  the  cylinder  in  inches ;  divide  this  quotient 
by  360  ;  finally  extract  the  cube  root  of  the  quotient.  The  result  is 
the  thickness  of  the  web  of  crank  at  paddle  shaft  centre  in  inches. 

Thus,  to  apply  the  rule  to  the  particular  example  which  we  have 
selected,  we  have 


THE   STEAM   ENGINE.  137 

48  =  length  of  crank  in  inches 
48 

"  UNI 


2304 
1-561  =  constant  multiplier. 

3596-5 
505-8 


4102-3 

64  =  diameter  of  cylinder. 
64 


4096 

•1235  =  constant  multiplier. 

505-8 
And  x/  4102-3  =  64-05  nearly. 

4096  =  square  of  diameter. 

360)  262348-5 

728-75 
And  V  782-75  =  9  nearly. 

Suppose  that  it  were  required  to  find  the  proper  diameter  for 
the  paddle  shaft  journal  of  an  engine  whose  stroke  is  8  feet  and 
diameter  of  cylinder  64  inches.  The  proper  diameter  of  the 
paddle  shaft  journal  in  this  case  is  14-06  inches. 

KULE. — To  find  the  diameter  of  the  paddle  shaft  journal. — Mul- 
tiply the  square  of  the  diameter  of  cylinder  in  inches  by  the  length 
of  the  crank  in  inches ;  extract  the  cube  root  of  the  product ; 
finally  multiply  the  result  by  -242.  The  final  product  is  the  diame- 
ter of  the  paddle  shaft  journal  in  inches. 

Thus,  to  apply  this  rule  to  the  particular  example  which  we  have 
before  selected,  we  have 

64  =  diameter  of  cylinder  in  inches. 
64 

4096 

48  =  length  of  crank  in  inches. 


196608 

and  ^196608  =  58-148 
but  58-148  x  -242  =  14-07  inches. 

Suppose  it  were  required  to  find  the  proper  length  of  the  paddle 
shaft  journal  for  an  engine  whose  stroke  is  8  feet,  and  diameter  of 
cylinder  64  inches.  The  proper  length  of  the  paddle  shaft  journal 
would  be,  in  this  case,  17*59  inches. 

The  following  rule  serves  for  engines  of  all  sizes : 

RULE. — To  find  the  length  of  the  paddle  shaft  journal. — Multiply 

the  square  of  the  diameter  of  the  cylinder  in  inches  by  the  length 

of  the  crank  in  inches ;  extract  the  cube  root  of  the  quotient ; 

multiply  the  result  by  -303.     The  product  is  the  length  of  the 

M2 


138          THE  PRACTICAL  MODEL  CALCULATOR. 

paddle  shaft  journal  in  inches.     (The  length  of  the  paddle  shaft 
journal  is  1J  times  the  diameter.) 

To  apply  this  rule  to  the  example  which  we  have  selected,  we  have 

64  =  diameter  of  cylinder  in  inches. 
64 


4096 

48  =  length  of  crank  in  inches. 

196608  

and  &  196608  =  58-148 
.-.  length  of  journal  =  58-148  x  -303  =  17-60  inches. 

We  shall  now  calculate  the  proper  dimensions  of  some  of  those 
parts  which  do  not  depend  upon  the  length  of  the  stroke.  Suppose 
it  were  required  to  find  the  proper  dimensions  of  the  respective  parts 
of  a  marine  engine  the  diameter  of  whose  cylinder  is  64  inches. 

Diameter  of  crank-pin  journal  ==  90-9  inches,  or  about  9  inches. 

Length  of  crank-pin  journal  =  10-18  inches,  or  nearly  10| 
inches. 

Breadth  of  the  eye  of  cross-head  =  2-64  inches,  or  between  2J 
and  2}  inches. 

Depth  of  the  eye  of  cross-head  =  18-37  inches,  or  very  nearly 
18J  inches. 

Diameter  of  the  journal  of  cross-head  =  5-5  inches,  or  5J  inches. 

Length  of  journal  of  cross-head  =  6-19  inches,  or  very  nearly  6| 
inches. 

Thickness  of  the  web  of  cross-head  at  middle  =  4-6  inches,  or 
somewhat  more  than  4J  inches. 

Breadth  of  web  of  cross-head  at  middle  =  17 '15  inches,  or 
between  17 A  and  17i  inches. 

Thickness  of  web  of  cross-head  at  journal  =  3*93  inches,  or 
very  nearly  4  inches. 

Breadth  of  web  of  cross-head  at  journal  =  6-46  inches,  or  nearly 
6J  inches. 

Diameter  of  piston  rod  =  6-4  inches,  or  6f  inches. 

Length  of  part  of  piston  rod  in  piston  =  12-8  inches,  or  12f 
inches. 

Major  diameter  of  part  of  piston  rod  in  cross-head  =  06-8  inches, 
or  nearly  6T^  inches. 

Minor  diameter  of  part  of  piston  rod  in  cross-head  =  5-76  inches, 
or  5}  inches. 

Major  diameter  of  part  of  piston  rod  in  piston  =  8-96  inches, 
or  nearly  9  inches. 

Minor  diameter  of  part  of  piston  rod  in  piston  =  7-36  inches, 
or  between  7J-  and  7J  inches. 

Depth  of  gibs  and  cutter  through  cross-head  =  6-72  inches,  or 
very  nearly  6f  inches. 

Thickness  of  gibs  and  cutter  through  cross-head  =  1*35  inches, 
or  between  1J  and  1J  inches. 


THE   STEAM  ENGINE.  139 

Depth  of  cutter  through  piston  =  5-45  inches,  or  nearly  5J  inches. 

Thickness  of  cutter  through  piston  =  2-24  inches,  or  nearly  2J 
inches. 

Diameter  of  connecting  rod  at  ends  =  6-08  inches,  or  nearly 
6-^  inches. 

Major  diameter  of  part  of  connecting  rod  in  cross-tail  =  6-27 
inches,  or  about  6J  inches. 

Minor  diameter  of  part  of  connecting  rod  in  cross-tail  =  5*76 
inches,  or  nearly  5f  inches. 

Breadth  of  butt  =  9 '98  inches,  or  very  nearly  10  inches. 

Thickness  of  butt  =  8  inches. 

Mean  thickness  of  strap  at  cutter  =  2*75  inches,  or  2f  inches. 

Mean  thickness  of  strap  above  cutter  =  2*06  inches,  or  some- 
what more  than  2  inches. 

Distance  of  cutter  from  end  of  strap  =  3-08  inches,  or  very 
nearly  3^  inches. 

Breadth  of  gibs  and  cutter  through  cross-tail  =  6-73  inches,  or 
very  nearly  6f  inches. 

Breadth  of  gibs  and  cutter  through  butt  =  7*04  inches,  or  some- 
what more  than  7  inches. 

Thickness  of  gibs  and  cutter  through  butt  =  1-84  inches,  or 
between  If  and  2  inches. 

These  results  are  calculated  from  the  following  rules,  which  give 
correct  results  for  all  sizes  of  engines. 

RULE  1.  To  find  the  diameter  of  crank-pin  journal. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -142.  The  result  is  the 
diameter  of  crank-pin  journal  in  inches. 

RULE  2.  To  find  the  length  of  crank-pin  journal. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  *16.  The  product  is  the 
length  of  the  crank-pin  journal  in  inches. 

RULE  3.  To  find  the  breadth  of  the  eye  of  cross-head. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -041.  The  product  is 
the  breadth  of  the  eye  in  inches. 

RULE  4.  To  find  the  depth  of  the  eye  of  cross-head. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -286.  The  product  is 
the  depth  of  the  eye  of  cross-head  in  inches. 

RULE  5.  To  find  the  diameter  of  the  journal  of  cross-head. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -086.  The  pro- 
duct is  the  diameter  of  the  journal  in  inches. 

RULE  6.  To  find  the  length  of  the  journal  of  cross-head. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  *097.  The  product 
is  the  length  of  the  journal  in  inches. 

RULE  7.  To  find  the  thickness  of  the  web  of  cross-head  at  middle. 
— Multiply  the  diameter  of  the  cylinder  in  inches  by  -072.  The 
product  is  the  thickness  of  the  web  of  cross-head  at  middle  in 
inches. 

RULE  8.  To  find  the  breadth  of  web  of  cross-head  at  middle. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -268.  The 
product  is  the  breadth  of  the  web  of  cross-head  at  middle  in  inches. 


140          THE  PRACTICAL  MODEL  CALCULATOR. 

RULE  9.  To  find  the  thickness  of  the  web  of  cross-head  at  journal. 
— Multiply  the  diameter  of  the  cylinder  in  inches  by  -061.  The 
product  is  the  thickness  of  the  web  of  cross-head  at  journal  in 
inches. 

RULE  10.  To  find  the  breadth  of  web  of  cross-head  at  journal. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  '101.  The 
product  is  the  breadth  of  the  web  of  cross-head  at  journal  in  inches. 

RULE  11.  To  find  the  diameter  of  the  piston  rod. — Divide  the 
diameter  of  the  cylinder  in  inches  by  10.  The  quotient  is  the 
diameter  of  the  piston  rod  in  inches. 

RULE  12.  To  find  the  length  of  the  part  of  the  piston  rod  in  the 
piston. — Divide  the  diameter  of  the  cylinder  in  inches  by  5.  The 
quotient  is  the  length  of  the  part  of  the  piston  rod  in  the  piston  in 
inches. 

RULE  13.  To  find  the  major  diameter  of  the  part  of  piston  rod  in 
cross-head. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•095.  The  product  is  the  major  diameter  of  the  part  of  piston  rod 
in  cross-head  in  inches. 

RULE  14.  To  find  the  minor  diameter  of  the  part  of -piston  rod  in 
cross-head. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -09. 
The  product  is  the  minor  diameter  of  the  part  of  piston  rod  in 
cross-head  in  inches. 

RULE  15.  To  find  the  major  diameter  of  the  part  of  piston  rod  in 
piston. — Multiply  the  diameter  of  the  cylinder  in  inches  by  *14. 
The  product  is  the  major  diameter  of  the  part  of  piston  rod  in 
piston  in  inches. 

RULE  16.  To  find  the  minor  diameter  of  the  part  of  piston  rod  in 
piston. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -115. 
The  product  is  the  minor  diameter  of  the  part  of  piston  rod  in 
piston. 

RULE  17.  To  find  the  depth  of  gibs  and  cutter  through  cross- 
head. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -105. 
The  product  is  the  depth  of  the  gibs  and  cutter  through  cross- 
head. 

RULE  18.  To  find  the  thickness  of  the  gibs  and  cutter  through 
cross-head. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•021.  The  product  is  the  thickness  of  the  gibs  and  cutter  through 
cross-head. 

RULE  19.  To  find  the  depth  of  cutter  through  piston. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -085.  The  product  is  the 
depth  of  the  cutter  through  piston  in  inches. 

RULE  20.  To  find  the  thickness  of  cutter  through  piston. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -035.  The  product 
is  the  thickness  of  cutter  through  piston  in  inches. 

RULE  21.  To  find  the  diameter  of  connecting  rod  at  ends. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -095.  The  product 
is  the  diameter  of  the  connecting  rod  at  ends  in  inches. 

RULE  22.  To  find  the  major  diameter  of  the  part  of  connecting 
rod  in  cross-tail. — Multiply  the  diameter  of  the  cylinder  in  inches 


THE   STEAM   ENGINE.  141 

by  '098.  The  product  is  the  major  diameter  of  the  part  of  con- 
necting rod  in  cross-tail. 

RULE  23.  To  find  the  minor  diameter  of  the  part  of  connecting 
rod  in  cross-tail. — Multiply  the  diameter  of  the  cylinder  in  inches 
by  -09.  The  product  is  the  minor  diameter  of  the  part  of  con- 
necting rod  in  cross-tail  in  inches. 

RULE  24.  To  find  the  breadth  of  butt. — Multiply  the  diameter 
of  the  cylinder  in  inches  by  -156.  The  product  is  the  breadth  of 
the  butt  in  inches. 

RULE  25.  To  find  the  thickness  of  the  butt. — Divide  the  diameter 
of  the  cylinder  in  inches  by  8.  The  quotient  is  the  thickness  of 
the  butt  in  inches. 

RULE  26.  To  find  the  mean  thickness  of  the  strap  at  cutter. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -043.  The  pro- 
duct is  the  mean  thickness  of  the  strap  at  cutter. 

RULE  27.  To  find  the  mean  thickness  of  the  strap  above  cutter. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -032.  The 
product  is  the  mean  thickness  of  the  strap  above  cutter. 

RULE  28.  To  find  the  distance  of  cutter  from  end  of  strap. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -048.  The 
product  is  the  distance  of  cutter  from  end  of  strap  in  inches. 

RULE  29.  To  find  the  breadth  of  the  gibs  and  cutter  through 
cross-tail. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•105.  The  product  is  the  breadth  of  the  gibs  and  cutter  through 
cross-tail. 

RULE  30.  To  find  the  breadth  of  the  gibs  and  cutter  through 
butt. — Multiply  the  diameter  of  the  cylinder  in  inches  by  •!!. 
The  product  is  the  breadth  of  the  gibs  and  cutter  through  butt  in 
inches. 

RULE  31.  To  find  the  thickness  of  the  gibs  and  cutter  through 
butt. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '029. 
The  product  is  the  thickness  of  the  gibs  and  cutter  through  butt 
in  inches. 

To  find  other  parts  of  the  engine  which  do  not  depend  upon  the 
stroke.  Suppose  it  were  required  to  find  the  thickness  of  the  small 
eye  of  crank  for  an  engine  the  diameter  of  whose  cylinder  is  64 
inches.  According  to  the  rule,  the  proper  thickness  of  the  small 
eye  of  crank  is  4-04  inches.  Again,  suppose  it  were  required  to 
find  the  length  of  the  small  eye  of  crank.  Hence,  according  to 
the  rule,  the  proper  length  of  the  small  eye  of  crank  is  11*94  inches. 
Again,  supposing  it  were  required  to  find  the  proper  thickness  of  the 
web  of  crank  at  pin  centre ;  that  is  to  say,  the  thickness  which  it 
would  have  if  continued  to  the  pin  centre.  According  to  the  rule, 
the  proper  thickness  for  the  web  of  crank  at  pin  centre  is  7*04  inches. 
Again,  suppose  it  were  required  to  find  the  breadth  of  the  web  of 
crank  at  pin  centre  ;  that  is  to  say,  the  breadth  which  it  would 
have  if  it  were  continued  to  the  pin  centre.  Hence,  according  to 
the  rule,  the  proper  breadth  of  the  web  of  crank  at  pin  centre  is 
10-24  inches. 


142          THE  PRACTICAL  MODEL  CALCULATOR. 

These  results  are  calculated  from  the  following  rules,  which  give 
the  proper  dimensions  for  engines  of  all  sizes : 

RULE  1.  To  find  the  breadth  of  the  small  eye  of  crank. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -063.  The  product  is 
the  proper  breadth  of  the  small  eye  of  crank  in  inches. 

RULE  2.  To  find  the  length  of  the  small  eye  of  crank. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -187.  The  product  is 
the  proper  length  of  the  small  eye  of  crank  in  inches. 

RULE  3.  To  find  the  thickness  of  the  web  of  crank  at  pin  centre. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -11.  The  pro- 
duct is  the  proper  thickness  of  the  web  of  crank  at  pin  centre  in 
inches. 

RULE  4.  To  find  the  breadth  of  the  web  of  crank  at  pin  centre. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  '16.  The  pro- 
duct is  the  proper  breadth  of  crank  at  pin  centre  in  inches. 

To  illustrate  the  use  of  the  succeeding  rules,  let  us  take  the  par- 
ticular example  of  an  engine  of  8  feet  stroke  and  64-inch  cylinder, 
and  let  us  suppose  that  the  length  of  the  connecting  rod  is  12 
feet,  and  the  side  rod  10  feet.  We  find  by  a  previous  rule  that 
the  diameter  of  the  connecting  rod  at  ends  is  6*08,  and  the  ratio 
between  the  diameters  at  middle  and  ends  of  a  connecting  rod, 
whose  length  is  12  feet,  is  1*504.  Hence,  the  proper  diameter  at 
middle  of  the  connecting  rod  =  6-08  X  1*504  inches  =  9-144 
inches.  And  again,  we  find  the  diameter  of  cylinder  side  rods  at 
ends,  for  the  particular  engine  which  we  have  selected,  is  4-10,  and 
the  ratio  between  the  diameters  at  middle  and  ends  of  cylinder 
side  rods,  whose  lengths  are  10  feet,  is  1*42.  Hence,  according  to 
the  rules,  the  proper  diameter  of  the  cylinder  side  rods  at  middle 
is  equal  to  4-1  X  1*42  inches  =  5-82  inches. 

To  find  some  of  those  parts  of  the  engine  which  do  not  depend 
upon  the  stroke.  Suppose  we  take  the  particular  example  of  an 
engine  the  diameter  of  whose  cylinder  is  64  inches.  We  find  from 
the  following  rules  that 

Diameter  of  cylinder  side  rods  at  ends  =  4-1  inches,  or  4-^ 
inches. 

Breadth  of  butt  =  4-93  inches,  or  very  nearly  5  inches. 

Thickness  of  butt  =  3-9  inches,  or  3$  inches. 

Mean  thickness  of  strap  at  cutter  =  2*06  inches,  or  a  little  more 
than  2  inches. 

Mean  thickness  of  strap  below  cutter  =  1*47  inches,  or  very 
nearly  1J  inches. 

Depths  of  gibs  and  cutter  =  5*12  inches,  or  a  little  more  than 
5^  inches. 

Thickness  of  gibs  and  cutter  =  1*03  inches,  or  a  little  more  than 
1  inch. 

Diameter  of  main  centre  journal  =  11*71  inches,  or  very  nearly 
11}  incnes. 

Length  of  main  centre  journal  =  17*6  inches,  or  17|  inches. 


THE   STEAM   ENGINE.  143 

Depth  of  eye  round  end  studs  of  lever  =  4*75  inches,  or  4|  inches. 

Thickness  of  eye  round  end  studs  of  lever  =  3-33  inches,  or  3J 
inches. 

Diameter  of  end  studs  of  lever  =  4-48  inches,  or  very  nearly  4J 
inches. 

Length  of  end  studs  of  lever  =  4*86  inches,  or  between  4f  and 
5  inches. 

Diameter  of  air-pump  studs  =  2-91  inches,  or  nearly  3  inches. 

Length  of  air-pump  studs  =  3-16  inches,  or  nearly  3|  inches. 

These  results  were  obtained  from  the  following  rules,  which  will 
be  found  to  give  the  proper  dimensions  for  all  sizes  of  engines. 

RULE  1.  To  find  the,  diameter  of  cylinder  side  rods  at  ends. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  *065.  The 
product  is  the  diameter  of  the  cylinder  side  rods  at  ends  in  inches. 

RULE  2.  To  find  the  breadth  of  butt  in  inches. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  '077.  The  product  is  the 
breadth  of  the  butt  in  inches. 

RULE  3.  To  find  the  thickness  of  the  butt. — Multiply  the  diameter 
of  the  cylinder  in  inches  by  "061.  The  product  is  the  thickness  of 
the  butt  in  inches. 

RULE  4.  To  find  the  mean  thickness  of  strap  at  cutter. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -032.  The  product 
is  the  mean  thickness  of  the  strap  at  cutter. 

RULE  5.  To  find  the  mean  thickness  of  strap  below  cutter. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -023.  The  product 
is  the  mean  thickness  of  strap  below  cutter  in  inches. 

RULE  6.  To  find  the  depth  of  gibs  and  cutter. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  '08.  The  product  is  the 
depth  of  the  gibs  and  cutter  in  inches. 

RULE  7.  To  find  the  thickness  of  gibs  and  cutter. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  '016.  The  product  is  the 
thickness  of  gibs  and  cutter  in  inches. 

RULE  8.  To  find  the  diameter  of  the  main  centre  journal. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -183.  The  product 
is  the  diameter  of  the  main  centre  journal  in  inches. 

RULE  9.  To  find  the  length  of  the  main  centre  journal. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -275.  The  product  is 
the  diameter  of  the  cylinder  in  inches. 

RULE  10.  To  find  the  depth  of  eye  round  end  studs  of  lever. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  '074.  The  pro- 
duct is  the  depth  of  the  eye  round  end  studs  of  lever  in  inches. 

RULE  11.  To  find  the  thickness  of  eye  round  end  studs  of  lever. 
— Multiply  the  diameter  of  the  cylinder  in  inches  by  -052.  The 
product  is  the  thickness  of  eye  round  end  studs  of  lever  in  inches. 

RULE  12.  To  find  the  diameter  of  the  end  studs  of  lever. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -07.  The  product 
is  the  diameter  of  the  end  studs  of  lever  in  inches. 

RULE  13.   To  find  the  length  of  the  end  studs  of  lever. — Multiply 


144          THE  PRACTICAL  MODEL  CALCULATOR. 

the  diameter  of  the  cylinder  in  inches  by  -076.  The  product  is  the 
length  of  the  end  studs  of  lever  in  inches. 

RULE  14.  To  find  the  diameter  of  the  air-pump  studs. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  '045.  The  product  is 
the  diameter  of  the  air-pump  studs  in  inches. 

RULE  15.  To  find  the  length  of  the  air-pump  studs. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -049.  The  product  is  the 
length  of  the  air-pump  studs  in  inches. 

The  next  rule  gives  the  proper  depth  in  inches  across  the  centre 
of  the  side  lever,  when,  as  is  generally  the  case,  the  side  lever  is 
of  cast  iron.  It  will  be  observed  that  the  depth  is  made  to  depend 
upon  the  diameter  of  the  cylinder  and  the  length  of  the  lever,  and 
not  at  all  upon  the  length  of  the  stroke,  except  indeed  in  so  far  as 
the  length  of  the  lever  may  depend  upon  the  length  of  the  stroke. 
Suppose  it  were  required  to  find  the  proper  depth  across  the  centre 
of  a  side  lever  whose  length  is  20  feet,  and  the  diameter  of  the 
cylinder  64  inches.  According  to  the  rule,  the  proper  depth 
across  the  centre  would  be  39 '26  inches. 

The  following  rule  will  give  the  proper  dimensions  for  any  size 
of  engine : 

RULE. — To  find  the  depth  across  the  centre  of  the  side  lever. — 
Multiply  the  length  of  the  side  lever  in  feet  by  '7423 ;  extract  the 
cube  root  of  the  product,  and  reserve  the  result  for  a  multiplier. 
Then  square  the  diameter  of  the  cylinder  in  inches ;  extract  the 
cube  root  of  the  result.  The  product  of  the  final  result  and  the 
reserved  multiplier  is  the  depth  of  the  side  lever  in  inches  across 
the  centre. 

Thus,  to  apply  this  rule  to  the  particular  example  which  we  have 
selected,  we  have 

20  =  length  of  side  lever  in  feet. 
•7423  =  constant  multiplier. 

14-846         

and  V  14-846  =  2-458  nearly. 
64  =  diameter  of  cylinder  in  inches. 
64 


4096 

and  ^"4096  =  16 

Hence  depth  at  centre  =  16  x  2-458  inches  =  39-33  inches,  or 
between  39J  and  39J  inches. 


The  next  set  of  rules  give  the  dimensions  of  several  of  the  parts 
of  the  air-pump  machinery  which  depend  upon  the  diameter  of  the 
cylinder  only.  To  illustrate  the  use  of  these  rules,  let  us  take  the 
particular  example  of  an  engine  the  diameter  of  whose  cylinder  is 
64  inches.  We  find  from  the  succeeding  rules  successively, 

Diameter  of  air-pump  =  38-4  inches,  or  38|  inches. 


THE   STEAM   ENGINE.  145 

Thickness  of  the  eye  of  air-pump  cross-head  =  1'58  inches,  or 
a  little  more  than  1J  inches. 

Depth  of  eye  of  air-pump  cross-head  =  11*01,  or  about  11  inches. 

Diameter  of  end  journals  of  air-pump  cross-head  =  3-29  inches, 
or  somewhat  more  than  3J  inches. 

Length  of  end  journals  of  air-pump  cross-head  =  3'7  inches,  or 
3T75  inches. 

Thickness  of  the  web  of  air-pump  cross-head  at  middle  =  2 '76 
inches,  or  a  little  more  than  2f  inches. 

Depth  of  web  of  air-pump  cross-head  at  middle  =  10*29  inches, 
or  somewhat  more  than  10J  inches. 

Thickness  of  web  of  air-pump  cross-head  at  journal  =  2-35 
inches,  or  about  2|  inches. 

Depth  of  web  of  air-pump  cross-head  at  journal  =  3-89  inches, 
or  about  3J  inches. 

Diameter  of  air-pump  piston  rod  when  made  of  copper  =  4*27 
inches,  or  about  4J  inches. 

Depth  of  gibs  and  cutter  through  air-pump  cross-head  =  4-04 
inches,  or  a  little  more  than  4  inches. 

Thickness  of  gibs  and  cutter  through  air-pump  cross-head  =  *81 
inches,  or  about  J  inch. 

Depth  of  cutter  through  piston  =  3-27  inches,  or  somewhat 
more  than  3J  inches. 

Thickness  of  cutter  through  piston  =  1'34  inches,  or  about  If 
inches. 

These  results  were  obtained  from  the  following  rules,  and  give 
the  proper  dimensions  for  all  sizes  of  engines  : 

RULE  1.  To  find  the  diameter  of  the  air-pump. — Multiply  the- 
diameter  of  the  cylinder  in  inches  by  *6.  The  product  is  the 
diameter  of  air-pump  in  inches. 

RULE  2.  To  find  the  thickness  of  the  eye  of  air-pump  cross-head. 
— Multiply  the  diameter  of  the  cylinder  in  inches  by  *025.  The 
product  is  the  thickness  of  the  eye  of  air-pump  cross-head  in  inches. 

RULE  3.  To  find  the  depth  of  eye  of  air-pump  cross-head. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  -171.  The  product 
is  the  depth  of  the  e*ye  of  air-pump  cross-head  in  inches. 

RULE  4.  To  find  the  diameter  of  the  journals  of  air-pump  cross- 
head. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -051. 
The  product  is  the  diameter  of  the  end  journals. 

RULE  5.  To  find  the  length  of  the  end  journals  for  air-pump 
cross-head. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•058.  The  product  is  the  length  of  the  air-pump  cross-head  jour- 
nals in  inches. 

RULE  6.  To  find  the  thickness  of  the  web  of  air-pump  cross-head 
at  middle. — Multiply  the  diameter  of  the  cylinder  in  inches  by  *043. 
The  product  is  the  thickness  at  middle  of  the  web  of  air-pump 
cross-head  in  inches. 

RULE  7.  To  find  the  depth  at  middle  of  the  web  of  air-pump  cross- 
head. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -161. 
N  10 


146          THE  PRACTICAL  MODEL  CALCULATOR. 

The  product  is  the  depth  at  middle  of  air-pump  cross-head  ID 
inches. 

RULE  8.  To  find  the  thickness  of  the  web  of  air-pump  cross- 
head  at  journals. — Multiply  the  diameter  of  the  cylinder  in  inches 
by  '037.  The  product  is  the  thickness  of  the  web  of  air-pump 
cross-head  at  journals  in  inches. 

RULE  9.  To  find  the  depth  of  the  air-pump  cross-head  web  at 
journals. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -061. 
The  product  is  the  depth  at  journals  of  the  web  of  air-pump  cross- 
heacL 

RULE  10.  To  find  the  diameter  of  the  air-pump  piston  rod  when 
of  copper. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•067.  The  product  is  the  diameter  of  the  air-pump  piston  rod, 
when  of  copper,  in  inches. 

RULE  11.  To  find  the  depth  of  gibs  and  cutter  through  air-pump 
cross-head. — Multiply  the  diameter  of  the  cylinder  in  inches  by 
•063.  The  product  is  the  depth  of  the  gibs  and  cutter  through  air- 
pump  cross-head  in  inches. 

RULE  12.  To  find  the  thickness  of  the  gibs  and  cutter  through 
air-pump  cross-head. — Multiply  the  diameter  of  the  cylinder  in 
inches  by  -013.  The  product  is  the  thickness  of  the  gibs  and 
cutter  in  inches. 

RULE  13.  To  find  the  depth  of  cutter  through  piston. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  *051.  The  product  is  the 
depth  of  the  cutter  through  piston  in  inches. 

RULE  14.  To  find  the  thickness  of  cutter  through  air-pump 
piston. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '021. 
The  product  is  the  thickness  of  the  cutter  through  air-pump  piston. 


The  next  seven  rules  give  the  dimensions  of  the  remaining  parts 
of  the  engine  which  do  not  depend  upon  the  stroke.  To  exemplify 
their  use,  suppose  it  were  required  to  find  the  corresponding  dimen- 
sions for  an  engine  the  diameter  of  whose  cylinder  is  64  inches. 
According  to  the  rule,  the  proper  diameter  of  the  air-pump  side 
rod  would  be  2'48  inches.  Hence,  according  to  the  rule,  the 
proper  breadth  of  butt  is  2-95  inches.  According  to  the  rule,  the 
proper  thickness  of  butt  is  2-35  inches.  According  to  the  rule, 
the  mean  thickness  of  strap  at  cutter  ought  to  be  1-24  inches. 
Hence,  according  to  the  rule,  the  mean  thickness  of  strap  below 
cutter  is  -91  inch.  According  to  the  rule,  the  proper  depth  for 
the  gibs  and  cutter  is  2-94  inches.  According  to  the  rule,  the 
proper  thickness  of  the  gibs  and  cutter  is  '63  inches. 

The  following  rules  give  the  correct  dimensions  for  all  sizes  of 
engines : 

RULE  1.  To  find  the  diameter  of  air-pump  side  rod  at  ends. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  -039.  The 
product  is  the  diameter  of  the  air-pump  side  rod  at  ends  in  inches. 

RULE  2.   To  find  the  breadth  of  butt  for  air-pump. — Multiply  the 


THE   STEAM   ENGINE.  147 

diameter  of  the  cylinder  in  inches  by  -046.  The  product  is  the 
breadth  of  butt  in  inches. 

RULE  3.  To  find  the  thickness  of  butt  for  air-pump. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -037.  The  product  is 
the  thickness  of  butt  for  air-pump  in  inches. 

RULE  4.  To  find  the  mean  thickness  of  strap  at  cutter. — Multiply 
the  diameter  of  the  cylinder  in  inches  by  -019.  The  product  is 
the  mean  thickness  of  strap  at  cutter  for  air-pump  in  inches. 

RULE  5.  To  find  the  mean  thickness  of  strap  below  cutter. — Mul- 
tiply the  diameter  of  the  cylinder  in  inches  by  0-14.  The  product 
is  the  mean  thickness  of  strap  below  cutter  in  inches. 

RULE  6.  To  find  the  depth  of  gibs  and  cutter  for  air-pump. — 
Multiply  the  diameter  of  the  cylinder  in  inches  by  0*48.  The 
product  is  the  depth  of  gibs  and  cutter  for  air-pump  in  inches. 

RULE  7.  To  find  the  thickness  of  gibs  and  cutter  for  air-pump. — 
Divide  the  diameter  of  the  cylinder  in  inches  by  100.  The 
quotient  is  the  proper  thickness  of  the  gibs  and  cutter  for  air-pump 
in  inches. 

With  regard  to  other  dimensions  made  to  depend  upon  the 
nominal  horse  power  of  the  engine : — Suppose  that  we  take  the 
particular  example  of  an  engine  whose  stroke  is  8  feet,  and  dia- 
meter of  cylinder  64  inches.  We  find  that  the  nominal  horse 
power  of  this  engine  is  nearly  175.  Hence  we  have  successively, 

Diameter  of  valve  shaft  at  journal  in  inches  =  4*85,  or  between 
4f  and  5  inches. 

Diameter  of  parallel  motion  shaft  at  journal  in  inches  =  3*91,  or 
very  nearly  4  inches. 

Diameter  of  valve  rod  in  inches  =  2*44,  or  about  2f  inches. 

Diameter  of  radius  rod  at  smallest  part  in  inches  =  1*97,  or 
very  nearly  2  inches. 

Area  of  eccentric  rod,  at  smallest  part,  in  square  inches  =  8 -37, 
or  about  8f  square  inches. 

Sectional  area  of  eccentric  hoop  in  square  inches  =  8'75,  or  8f 
square  inches. 

Diameter  of  eccentric  pin  in  inches  =  2-24,  or  2J  inches. 

Breadth  of  valve  lever  for  eccentric  pin  at  eye  in  inches  =  5*7, 
or  very  nearly  5f  inches. 

Thickness  of  valve  lever  for  eccentric  pin  at  eye  in  inches  =  3. 

Breadth  of  parallel  motion  crank  at  eye  =  4-2  inches,  or  very 
nearly  4J  inches. 

Thickness  of  parallel  motion  crank  at  eye  =1/76  inches,  or 
about  If  inches. 

To  find  the  area  in  square  inches  of  each  steam  port.  Suppose 
it  were  required  to  find  the  area  of  each  steam  port  for  an  engine 
whose  stroke  is  8  feet,  and  diameter  of  cylinder  64  inches.  Accord- 
ing to  the  rule,  the  area  of  each  steam  port  would  be  202*26  square 
inches. 

With  regard  to  the  rule,  we  may  remark  that  the  area  of  the 


148  THE   PRACTICAL   MODEL    CALCULATOR. 

steain  port  ought  to  depend  principally  upon  the  cubical  content  of 
the  cylinder,  which  again  depends  entirely  upon  the  product  of  the 
square  of  the  diameter  of  the  cylinder  and  the  length  of  the  stroke 
of  the  engine.  It  is  well  known,  however,  that  the  quantity  of 
steam  admitted  by  a  small  hole  does  not  bear  so  great  a  proportion 
to  the  quantity  admitted  by  a  larger  one,  as  the  area  of  the  one  does 
to  the  area  of  the  other ;  and  a  certain  allowance  ought  to  be  made 
for  this.  In  the  absence  of  correct  theoretical  information  on  this 
point,  we  have  attempted  to  make  a  proper  allowance  by  supplying 
a  constant ;  but  of  course  this  plan  ought  only  to  be  regarded  as 
an  approximation.  Our  rule  is  as  follows : 

RULE. — To  find  the  area  of  each  steam  port. — Multiply  the 
square  of  the  diameter  of  the  cylinder  in  inches  by  the  length  of 
the  stroke  in  feet ;  multiply  this  product  by  11 ;  divide  the  last 
product  by  1800  ;  and,  finally,  to  the  quotient  add  8.  The  result 
is  the  area  of  each  steam  port  in  square  inches. 

To  show  the  use  of  this  rule,  we  shall  apply  it  to  a  particular 
example.  We  shall  apply  it  to  an  engine  whose  stroke  is  6  feet, 
and  diameter  of  cylinder  30  inches.  Then,  according  to  the  rule, 
we  have 

30  =  diameter  of  the  cylinder  in  inches. 
JO 

900  =  square  of  diameter. 
6  =  length  of  stroke  in  feet. 

5400 
11 


59400  -f- 1800  =  33 

J8  =  constant  to  be  added. 

41  =  area  of  steam  port  in  square  inches. 

When  the  length  of  the  opening  of  steam  port  is  from  any  cir- 
cumstance found,  the  corresponding  depth  in  inches  may  be  found, 
by  dividing  the  number  corresponding  to  the  particular  engine,  by 
the  given  length  in  inches :  conversely,  the  length  may  be  found, 
when  for  some  reason  or  other  the  depth  is  fixed,  by  dividing  the 
number  corresponding  to  the  particular  engine,  by  the  given  depth 
in  inches :  the  quotient  is  the  length  in  inches. 

The  next  rule  is  useful  for  determining  the  diameter  of  the  steam 
pipe  branching  off  to  any  particular  engine.  Suppose  it  were 
required  to  find  the  diameter  of  the  branch  steam  pipe  for  an 
engine  whose  stroke  is  8  feet,  and  diameter  of  cylinder  64  inches. 
According  to  the  rule,  the  proper  diameter  of  the  steam  pipe 
would  be  13'16  inches. 

The  following  rule  will  be  found  to  give  the  proper  diameter  of 
steam  pipe  for  all  sizes  of  engines. 

RULE. — To  find  the  diameter  of  branch  steam  pipe. — Multiply 
together  the  square  of  the  diameter  of  the  cylinder  in  inches,  the 


THE   STEAM   ENGINE.  149 

length  of  the  stroke  in  feet,  and  -00498;  to  the  product  add  10-2, 
and  extract  the  square  root  of  the  sum.  The  result  is  the  diameter 
of  the  steam  pipe  in  inches. 

To  exemplify  the  use  of  this  rule  we  shall  take  an  engine  whose 
stroke  is  8  feet,  and  diameter  of  cylinder  64  inches.  In  this  case 
we  have  as  follows  : — 

64  =  diameter  of  cylinder  in  inches. 

64 

4096  =  square  of  diameter. 
8  =  length  of  stroke  in  feet. 


32768 

•00498  =  constant  multiplier. 

163-18 
10-2          =  constant  to  be  added. 

173-38 
and  V  173-38  =  13-16. 

To  find  the  diameter  of  the  pipes  connected  with  the  engine. 
They  are  made  to  depend  upon  the  nominal  horse  power  of  the 
engine.  Suppose  it  were  required  to  apply  this  rule  to  determine 
the  size  of  the  pipes  for  two  marine  engines,  whose  strokes  are 
each  8  feet,  and  diameters  of  cylinder  each  64  inches.  We  find 
the  nominal  horse  power  of  each  of  these  engines  to  be  174-3. 
Hence,  according  to  the  rules,  we  have  in  succession, 

Diameter  of  waste  water  pipe  =  15-87  inches,  or  between  15f 

and  16  inches. 

Area  of  foot-valve  passage  =  323  square  inches. 
Area  of  injection  pipe  =  14-88  square  inches. 
If  the  injection  pipe  be  cylindrical,  then  by  referring  to  the 
table  of  areas  of  circles,  we  see  that  its  diameter  would  be  about 
4f  inches. 

Diameter  of  feed  pipe  =  4-12  inches,  or  between  4  and  4J 

inches. 
Diameter  of  waste  steam  pipe  =  12-17  inches,  or  nearly  12 J 

inches. 
Diameter  of  safety  valve, 

When  one  is  used  =14*05  inches. 
When  two  are  used  =  9-94  inches. 
When  three  are  used  =  8-12  inches. 
When  four  are  used  =  7-04  inches. 

These  results  were  obtained  from  the  following  rules,  which  will 
give  the  correct  dimensions  for  all  sizes  of  engines. 

RULE  1.  To  find  the  diameter  of  waste  water  pipe. — Multiply 
the  square  root  of  the  nominal  horse  power  of  the  engine  by  1-2. 
The  product  is  the  diameter  of  the  waste  water  pipe  in  inches. 

RULE  2.   To  find  the  area  of  foot-valve  passage. — Multiply  the 

N  2 


150  THE  PRACTICAL  MODEL  CALCULATOR. 

nominal  horse  power  of  the  engine  by  9  ;  divide  the  product  by  5  ; 
add  8  to  the  quotient.  The  sum  is  the  area  of  foot-valve  passage 
in  square  inches. 

RULE  3.  To  find  the  area  of  injection  pipe. — Multiply  the  nomi- 
nal horse  power  of  the  engine  by  -069  ;  to  the  product  add  2*81. 
The  sum  is  the  area  of  the  injection  pipe  in  square  inches. 

RULE  4.  To  find  the  diameter  of  feed  pipe. — Multiply  the  nomi- 
nal horse  power  of  the  engine  by  -04 ;  to  the  product  add  3  ;  extract 
the  square  root  of  the  sum.  The  result  is  the  diameter  of  the  feed 
pipe  in  inches. 

RULE  5.  To  find  the  diameter  of  waste  steam  pipe. — Multiply 
the  collective  nominal  horse  power  of  the  engines  by  -375 ;  to  the 
product  add  16*875 ;  extract  the  square  root  of  the  sum.  The 
final  result  is  the  diameter  of  the  waste  steam  pipe  in  inches. 

RULE  6.  To  find  the  diameter  of  the  safety  valve  when  only  one 
is  used. — To  one-half  the  collective  nominal  horse  power  of  the 
engines  add  22*5  ;  extract  the  square  root  of  the  sum.  The  result 
is  the  diameter  of  the  safety  valve  when  only  one  is  used. 

RULE  7.  To  find  the  diameter  of  the  safety  valve  when  tivo  are 
used. — Multiply  the  collective  nominal  horse  power  of  the  engines 
by  -25  ;  to  the  product  add  11-25 ;  extract  the  square  root  of  the 
sum.  The  result  is  the  diameter  of  the  safety  valve  when  two 
are  used. 

RULE  8.  To  find  the  diameter  of  the  safety  valve  when  three  are 
used. — To  one-sixth  of  the  collective  nominal  horse  power  of  the 
engines  add  7*5 ;  extract  the  square  root  of  the  sum.  The  result 
is  the  diameter  of  the  safety  valve  where  three  are  used. 

RULE  9.  To  find  the  diameter  of  the  safety  valve  when  four  are 
used. — Multiply  the  collective  nominal  horse  power  of  the  engines 
by  '125  ;  to  the  product  add  5-625 ;  extract  the  square  root  of  the 
sum.  The  result  is  the  diameter  of  the  safety  valve  when  four 
are  used. 

Another  rule  for  safety  valves,  and  a  preferable  one  for  low 
pressures,  is  to  allow  '8  of  a  circular  inch  of  area  per  nominal 
horse  power. 

The  next  rule  is  for  determining  the  depth  across  the  web  of  the 
main  beam  of  a  land  engine.  Suppose  we  wished  to  find  the  proper 
depth  at  the  centre  of  the  main  beam  of  a  land  engine  whose  main 
beam  is  16  feet  long,  and  diameter  of  cylinder  64  inches.  Accord- 
ing to  the  rule,  the  proper  depth  of  the  web  across  the  centre  is 
46-17  inches.  This  rule  gives  correct  dimensions  for  all  sizes  of 
engines. 

RULE. —  To  find  the  depth  of  the  web  at  the  centre  of  the  main 
beam  of  a  land  engine. — Multiply  together  the  square  of  the  di- 
ameter of  the  cylinder  in  inches,  half  the  length  of  the  main  beam 
in  feet,  and  the  number  3 ;  extract  the  cube  root  of  the  product. 
The  result  is  the  proper  depth  of  the  web  of  the  main  beam  across 
the  centre  in  inches,  when  the  main  beam  is  constructed  of  cast 
iron. 


THE    STEAM    ENGINE.  151 

To  illustrate  this  rule  we  shall  take  the  particular  example  of  an 
engine  whose  main  beam  is  20  feet  long,  and  the  diameter  of  the 
cylinder  64  inches.  In  this  case  we  have 

64  =  diameter  of  cylinder  in  inches. 
64 

4096  =  square  of  the  diameter. 

10  =  J  length  of  main  beam  in  feet. 


40960 

3  =  constant  multiplier. 


122880 


0               0  122880(49-714  =  ^122880 

4  Ii6  64 

4  16  58880 

4  32  53649 

8  4800  5231 

4  1161  5112 

120  5961  119 

9  1242  74 

129  7203  35 
_9              10 

138  730 
9  10 


147  741 

To  find  the  depth  of  the  main  beam  across  the  ends.  Suppose 
it  were  required  to  find  the  depth  at  ends  of  a  cast-iron  main  beam 
whose  length  is  20  feet,  when  the  diameter  of  the  cylinder  is  64 
inches.  The  proper  depth  will  be  19-89  inches. 

The  following  rule  gives  the  proper  dimensions  for  all  sizes  of 
engines. 

RULE. —  To  find  the  depth  of  main  beam  at  ends. — Multiply  to- 
gether the  square  of  the  diameter  of  the  cylinder  in  inches,  half 
the  length  of  the  main  beam  in  feet,  and  the  number  '192  ;  extract 
the  cube  root  of  the  product.  The  result  is  the  depth  in  inches  of 
the  main  beam  at  ends,  when  of  cast  iron. 

To  illustrate  this  rule,  let  us  apply  it  to  the  particular  example 
of  an  engine  whose  main  beam  is  20  feet  long,  and  the  diameter 
of  the  cylinder  64  inches.     In  this  case  we  have  as  follows : 
64  =  diameter  of  cylinder  in  inches. 
64 

4096  =  square  of  diameter  of  cylinder. 
10  =  i  length  of  main  beam  in  feet. 


40960 
•192  =  constant  multiplier. 

7864-32 


152          THE  PRACTICAL  MODEL  CALCULATOR. 

0  0  7864-32  ( 19-89  = 


1  1  6864 

1.  2_  5859 

IT  300  1005 

1  351  898 

30  651  107 

__9  432 

"39  1083 

_9  4         . 

48  112 

_9  4 

57  116 

so  that,  according  to  the  rule,  the  depth  at  ends  is  nearly  20  inches. 

To  find  the  dimensions  of  the  feed-pump  in  cubic  inches.  Sup- 
pose we  take  the  particular  example  of  an  engine  whose  stroke  is 
8  feet,  and  diameter  of  cylinder  64  inches.  The  proper  content  of 
the  feed-pump  would.be  1093-36  cubic  inches.  Suppose,  now, 
that  the  cold-water  pump  was  suspended  from  the  main  beam  at  a 
fourth  of  the  distance  between  the  centre  and  the  end,  so  that  its 
stroke  would  be  2  feet,  or  24  inches.  In  this  case  the  area  of  the 
pump  would  be  equal  to  1093-36  -H  24  =  45-556  square  inches ; 
so  that  we  conclude  that  the  diameter  is  between  7J  and  7f  inches. 
Conversely,  suppose  that  it  was  wished  to  find  the  stroke  of  the 
pump  when  the  diameter  was  5  inches.  We  find  the  area  of  the 
pump  to  be  19-635  square  inches ;  so  that  the  stroke  of  the  feed- 
pump must  be  equal  to  1093-36  -r- 19-635  =  55-69  inches,  or  very 
nearly  55f  inches. 

This  rule  will  be  found  to  give  correct  dimensions  for  all  sizes 
of  engines : 

RULE. — To  find  the  content  of  the  feed-pump. — Multiply  the 
square  of  the  diameter  of  the  cylinder  in  inches  by  the  length  of 
the  stroke  in  feet ;  divide  the  product'  by  30.  The  quotient  is  the 
content  of  the  feed-pump  in  cubic  inches. 

Thus,  for  an  engine  whose  stroke  is  6  feet,  and  diameter  of  cylin- 
der 50  inches,  we  have, 

50  =  diameter  of  cylinder. 
50 

2500  =  square  of  the  diameter  of  the  cylinder. 
6  =  length  of  stroke  in  feet. 


30)15000 

500  =  content  of  feed-pump  in  cubic  inches. 
To  determine  the  content  of  the  cold-water  pump  in  cubic  feet. 
To  illustrate  this,  suppose  we  take  the  particular  example  of  an  en- 


THE   STEAM   ENGINE.  153 

gine  whose  stroke  is  8  feet,  and  diameter  of  cylinder  64  inches. 
Suppose,  now,  the  stroke  of  the  pump  to  be  5  feet,  then  the  area 
equal  to  7*45  -4-  5  =  1*49  square  feet  =  214-56  square  inches ; 
we  see  that  the  diameter  of  the  pump  is  about  16J  inches.  Again, 
suppose  that  the  diameter  of  the  cold-water  pump  was  20  inches, 
and  that  it  was  required  to  find  the  length  of  its  stroke.  The  area 
of  the  pump  is  314*16  square  inches,  or  314-16  -s-  144  =  2-18 
square  feet ;  so  that  the  stroke  of  the  pump  is  equal  to  7*45  -*- 
2-18  =  3-42  feet. 

The  content  is  calculated  from  the  following  rule,  which  will  be 
found  to  give  correct  dimensions  for  all  sizes  of  engines  : 

RULE. — To  find  the  content  of  the  cold-water  pump. — Multiply 
the  square  of  the  diameter  of  the  cylinder  in  inches  by  the  length 
of  the  stroke  in  feet ;  divide  the  product  by  4400.  The  quotient 
is  the  content  of  the  cold-water  pump  in  cubic  feet. 

To  explain  this  rule,  we  shall  take  the  particular  example  of  an 
engine  whose  stroke  is  5J  feet,  and  diameter  of  cylinder  60  inches. 
In  this  case  we  have  in  succession, 

60  =  diameter  of  cylinder  in  inches. 
60 


3600  =  square  of  the  diameter  of  cylinder^. 
5J  =  length  of  stroke  in  feet. 


4400)19800 

4-5  =  content  of  cold  water  pump  in  cubic  feet. 

To  determine  the  proper  thickness  of  the  large  eye  of  crank  for 
fly-wheel  shaft  when  the  crank  is  of  cast  iron.  The  crank  is  some- 
times cast  on  the  shaft,  and  of  course  the  thickness  of  the  large 
eye  is  not  then  so  great  as  when  the  crank  is  only  keyed  on  the 
shaft,  or  rather  there  is  then  no  large  eye  at  all.  To  illustrate  the 
use  of  this  rule,  we  shall  apply  it  to  the  particular  example  of  an 
engine  whose  stroke  is  8  feet,  and  diameter  of  cylinder  64  inches. 
Hence,  according  to  the  rule,  the  proper  thickness  of  the  large  eye 
of  crank  when  of  cast  iron  is  8*07  inches.  For  a  marine  engine 
of  8  feet  stroke  and  64  inch  cylinder,  the  thickness  of  the  large 
eye  of  crank  is  about  5}  inches.  The  difference  is  thus  about  2J 
inches,  which  is  an  allowance  for  the  inferiority  of  cast  iron  to 
malleable  iron. 

The  following  rule  will  be  found  to  give  correct  dimensions  for 
all  sizes  of  engines : 

RULE. — To  find  the  thickness  of  the  large  eye  of  crank  for  fly- 
wheel shaft  when  of  cast  iron. — Multiply  the  square  of  the  length 
of  the  crank  in  inches  by  1-561,  and  then  multiply  the  square  of  the 
diameter  of  the  cylinder  in  inches  by  -1235  ;  multiply  the  sum  of 
these  products  by  the  square  of  the  diameter  of  cylinder  in  inches ; 
divide  this  product  by  666-283 ;  divide  this  quotient  by  the  length 
of  the  crank  in  inches  ;  finally  extract  the  cube  root  of  the  quotient. 


154          THE  PRACTICAL  MODEL  CALCULATOR. 

The  result  is  the  proper  thickness  of  the  large  eye  of  crank  for 
fly-wheel  shaft  in  inches,  when  of  cast  iron. 

As  this  rule  is  rather  complicated,  we  shall  show  its  application 
to  the  particular  example  already  selected. 

48  =  length  of  crank  in  inches. 
-     48          g 


2304  =  square  of  length  of  crank  in  inches. 
1*561  =•  constant  multiplier. 

359^5 

64  =  diameter  of  cylinder  in  inches. 
64 

4096  =  square  of  the  diameter  of  cylinder. 
•1235  =  constant  multiplier. 

505-8 
3596-5 

4102-3  =  sum  of  products. 
4096  =  square  of  the  diameter  of  cylinder. 

666-283)  16803020-8 
length  of  crank=48)  25219-045 

525-397 
and  ^525-397  =  8-07  nearly. 

To  find  the  breadth  of  the  web  of  crank  at  the  centre  of  the  fly- 
wheel shaft,  that  is  to  say,  the  breadth  which  it  would  have  if  it- 
were  continued  to  the  centre  of  the  fly-wheel  shaft.  Suppose  it 
were  required  to  find  the  breadth  of  the  crank  at  the  centre  of  the 
fly-wheel  shaft  for  an  engine  whose  stroke  is  8  feet,  and  diameter 
of  cylinder  64  inches.  According  to  the  rule,  the  proper  breadth 
is  22-49  inches.  According  to  a  former  rule,  the  breadth  of  the 
web  of  a  cast  iron  crank  of  an  engine  whose  stroke  is  8  feet,  and 
diameter  of  cylinder  64  inches,  is  about  18  inches.  The  difference 
between  these  two  is  about  4J  inches ;  which  is  not  too  great  an 
allowance  for  the  inferiority  of  the  cast  iron. 

The  following  rule  will  be  found  to  give  correct  dimensions  for 
all  sizes  of  engines : 

RULE. —  To  find  the  breadth  of  the  web  of  crank  at  fly-wheel  shaft^ 
when  of  east  iron. — Multiply  the  square  of  the  length  of  the  crank 
in  inches  by  1'561,  and  then  multiply  the  square  of  the  diameter 
of  the  cylinder  in  inches  by  -1235 ;  multiply  the  square  root  of  the 
sum  of  these  products  by  the  square  of  the  diameter  of  the  cylinder 
in  inches ;  divide  the  product  by  23-04,  and  finally  extract  the 
cube  root  of  the  quotient.  The  final  result  is  the  breadth  of  the 
crank  at  the  centre  of  the  fly-wheel  shaft,  when  the  crank  is  of 
cast  iron. 

As  this  rule  is  rather  complicated,  we  shall  illustrate  it  by  show- 


THE   STEAM   ENGINE.  155 

ing  its  application  to  the  particular  example  of  an  engine  whose 
stroke  is  8  feet,  and  diameter  of  cylinder  64  inches. 

64  =  diameter  of  cylinder  in  inches. 
64 


4096  =  square  of  the  diameter  of  cylinder. 
•1235  =  constant  multiplier. 

50^8 

48  =  length  of  crank  in  inches. 

48 


2304  =  square  of  the  length  of  crank. 
1-561  =  constant  multiplier. 

3596-5 
505-8 

4102-3  =  sum  of  products. 
^4102-3  =  64-05  nearly. 

4096  =  square  of  the  diameter  of 
constant  divisor  =  23-04)  262348-5  [cylinder. 

11386-66  nearly. 

and  ^11386-66  =  22-49. 

• 

To  determine  the  thickness  of  the  web  of  crank  at  the  centre  of 
the  fly-wheel  shaft ;  that  is  to  say,  the  thickness  which  it  would 
have  if  it  were  continued  so  far.  Suppose  it  were  required  to  find 
the  thickness  of  web  of  crank  at  the  centre  of  fly-wheel  shaft  of 
an  engine  whose  stroke  is  8  feet,  and  diameter  of  cylinder  64 
inches.  According  to  the  rule,  the  proper  thickness  would  be 
11-26  inches.  "The  proper  thickness  of  web  at  centre  of  paddle 
shaft  for  a  marine  engine  whose  stroke  is  8  feet,  and  diameter  of 
cylinder  64  inches,  is  nearly  9  inches.  The  difference  between  the 
two  thicknesses  is  about  2J  inches,  which  is  not  too  great  an  allow- 
ance for  the  inferiority  of  cast  iron  to  malleable  iron. 

The  following  rule  will  be  found  to  give  correct  dimensions  for 
all  sizes  of  engines  : 

RULE. — To  find  the  thickness  of  the  web  of  crank  at  centre  of 
fly-wheel  shaft,  when  of  cast  iron. — Multiply  the  square  of  the 
length  of  the  crank  in  inches  by  1-561,  and  then  multiply  the 
square  of  the  diameter  of  the  cylinder  in  inches  by  -1235  ;  multi- 
ply the  square  root  of  the  sum  of  these  products  by  the  square  of 
the  diameter  of  the  cylinder  in  inches ;  divide  this  product  by 
184-32 ;  finally  extract  the  cube  root  of  the  quotient.  The  result 
is  the  thickness  of  the  web  of  crank  at  the  centre  of  the  fly-wheel 
shaft  when  of  cast  iron,  in  inches. 

As  this  rule  is  rather  complicated,  we  shall  illustrate  it  by  apply  • 
ing  it  to  the  particular  engine  which  we  have  already  selected. 


156          THE  PRACTICAL  MODEL  CALCULATOR. 

48  =  length  of  crank  in  inches. 

48 


2304  =  square  of  length  of  crank. 
1*561  =  constant  multiplier. 

3596^5 

64  =  diameter  of  cylinder  in  inches. 
64 


4096  =  square  of  the  diameter  of  cylinder. 
•1235  =  constant  multiplier. 

505-8 
3596-5 

4102-3  =  sum  of  products. 


and  ^  4102-3  =  64-05  nearly. 

4096  =  square  of  diameter. 
Constant  divisor  =  184-32)  26234'M 

1423-33 

and  ^1423-33  =11-24 

To  find  the  proper  diameter  of  the  fly-wheel  shaft  at  its  smallest 
part,  when,  as  is  usually  the  case,  it  is  of  cast  iron.  Suppose  it 
were  required  to  find  the  diameter  of  the  fly-wheel  shaft  for  an 
engine  whose  stroke  is  8  feet,  and  diameter  of  cylinder  64  inches. 
According  to  the  rule,  the  diameter  would  be  17'59  inches.  It  is 
obvious  enough  that  the  fly-wheel  shaft  stands  in  much  the  same 
relation  to  the  land  engine,  as  the  paddle  shaft  does  to  the  marine 
engine.  According  to  a  former  rule,  the  diameter  of  the  paddle 
shaft  journal  of  a  marine  engine  whose  stroke  is  8  feet,  and  dia- 
meter of  cylinder  64  inches,  is  about  14  inches.  The  difference 
betwixt  the  diameter  of  the  paddle  shaft  for  the  marine  engine, 
and  the  diameter  of  the  fly-wheel  shaft  for  the  corresponding  land 
engine  is  about  3J  inches.  This  will  be  found  to  be  a  very  proper 
allowance  for  the  different  circumstances  connected  with  the  land 
engine. 

The  following  rule  will  be  found  to  give  correct  dimensions  for  all 
sizes  of  engines. 

RULE. — To  find  the  diameter  of  the  fly-wheel  shaft  at  smallest 
party  when  it  is  of  cast  iron. — Multiply  the  square  of  the  diameter 
of  the  cylinder  in  inches  by  the  length  of  the  crank  in  inches ; 
extract  the  cube  root  of  the  product ;  finally  multiply  the  result 
by  -3025.  The  result  is  the  diameter  of  the  fly-wheel  shaft  at 
smallest  part  in  inches. 

We  shall  illustrate  this  rule  by  applying  it  to  the  particular 
engine  which  we  have  already  selected. 


THE   STEAM   ENGINE.  157 

64  =  diameter  of  cylinder  in  inches. 
64 


4096  =  square  of  the  diameter. 
48  =  length  of  crank  in  inches. 

196608 

0                 0  196608  (58-15  = 

5               _25  125 

.  5                25  71608 

_5                50  70112 

10                7500  1496 

_5                1264  1011 

150                8764  485 
__8                1328 

158 

8 


166  1011 

__8  _2 

174  1013 

and  58-15  x  -3025  =  1759 
which  agrees  with  the  number  given  by  a  former  rule. 

To  determine  the  sectional  area  of  the  fly-wheel  rim  when  of 
cast  iron.  Suppose  it  were  required  to  find  the  sectional  area  of 
the  rim  of  a  fly-wheel  for  an  engine  whose  stroke  is  8  feet,  and 
diameter  of  cylinder  64  inches,  the  diameter  of  the  fly-wheel  itself 
being  30  feet.  According  to  the  rule,  the  sectional  area  of  the 
rim  in  square  inches  =  146-4  X  -813  =  119-02.  We  may  remark 
that  this  calculation  has  been  made  on  the  supposition  that  the  fly- 
wheel is  so  connected  with  the  engine,  as  to  make  exactly  one  revo- 
lution for  each  double  stroke  of  the  piston.  If  the  fly-wheel  is  so 
connected  with  the  engine  as  to  make  more  than  one  revolution  for 
each  double  •  stroke,  then  the  rim  does  not  need  to  be  so  heavy  as 
we  make  it.  If,  on  the  contrary,  the  fly-wheel  does  not  make  a 
complete  revolution  for  each  double  stroke  of  the  engine,  then  it 
ought  to  be  heavier  than  this  rule  makes  it. 

RULE. —  To  find  the  sectional  area  of  the  rim  of  the  fly-wheel 
when  of  cast  iron. — Multiply  together  the  square  of  the  diameter 
of  the  cylinder  in  inches,  the  square  of  the  length  of  the  stroke 
in  feet,  the  cube  root  of  the  length  of  the  stroke  in  feet,  and  6*125 ; 
divide  the  final  product  by  the  cube  of  the  diameter  of  the  fly-wheel 
in  feet.  The  quotient  is  the  sectional  area  of  the  rim  of  fly-wheel 
in  square  inches,  provided  it  is  of  cast  iron. 

As  this  rule  is  rather  complicated,  we  shall  endeavour  to  illustrate 
it  by  showing  its  application  to  a  particular  engine.  We  shall 
apply  the  rule  to  determine  the  sectional  area  of  the  rim  of  fly- 


158  THE  PRACTICAL  MODEL  CALCULATOR. 

wheel  for  an  engine  whose  stroke  is  8  feet,  diameter  of  cylinder  50 
inches;  the  diameter  of  the  fly-wheel  being  20  feet.  For  this 
engine  we  have  as  follows : 

2500  =  square  of  diameter  of  cylinder. 
64  =  square  of  the  length  of  stroke. 


160000 

2  =  cube  root  of  the  length  of  stroke. 

320000 
6-125  =  constant  multiplier. 

1960000 

therefore   sectional   area  in  square  ijiches  =  1960000  -5-  203  = 
1960000  -r-  8000  =  1960  -*-  8  =  245. 

In  the  following  formulas  we  denote  the  diameter  of  the  cylinder 
in  inches  by  D,  the  length  of  the  crank  in  inches  by  R,  the  length 
of  the  stroke  in  feet,  and  the  nominal  horse  power  of  the  engine 
by  H.R 

MARINE   ENGINES. — DIMENSIONS  OF  SEVERAL  OF  THE  PARTS  OF  THE 
SIDE  LEVER. 

Depth  of  eye  round  end  studs  of  lever  =  *074  X  D. 
,  Thickness  of  eye  round  end  studs  of  lever  =  *052  X  D. 
Diameter  of  end  studs,  in  inches  =  '07  X  D. 
Length  of  end  studs,  in  inches  =  '076  X  D. 
Diameter  of  air-pump  studs,  in  inches  =  -045  X  D. 
Length  of  air-pump  studs,  in  inches  =  -049  X  D. 

2. 

Depth  of  cast  iron  side  lever  across  centre,  in  inches  =  D3  X 
{•7423  X  length  of  lever  in  feet}*. 

MARINE   ENGINE. — DIMENSIONS    OF    SEVERAL   PARTS   OF   AIR-PUMP 
CROSS-HEAD. 

Diameter  of  air-pump,  in  inches  =  -6  x  D. 
Thickness  of  eye  for  air-pump  rod,  in  inches  =  *025  X  D. 
Depth  of  eye  for  air-pump  rod,  in  inches  =  *171  X  D. 
Diameter  of  end  journals,  in  inches  =  *051  X  D. 
Length  of  end  journals,  in  inches  =  '058  X  D. 
Thickness  of  web  at  middle,  in  inches  =  '043  X  D. 
Depth  of  web  at  middle,  in  inches  =  -161  X  D. 
Thickness  of  web  at  journal  =  -037  X  D. 
Depth  of  web  at  journal  =  -061  X  D. 

MARINE   ENGINE. — DIMENSIONS     OF    THE   PARTS    OF    AIR-PUMP 
PISTON-ROD. 

Diameter  of  air-pump  piston-rod,  when  of  copper,  in  inches  = 
•067  x  D. 

Depth  of  gibs  and  cutter  through  cross-head,  in  inches  = 
•063  x  D. 


THE   STEAM   ENGINE.  159 

Thickness  of  gibs  and  cutter  through  cross-head,  in  inches  = 
•013  x  D. 

Depth  of  cutter  through  piston,  in  inches  =  *051  X  D. 
Thickness  of  cutter  through  piston,  in  inches  =  "021  X  D. 

MARINE  ENGINE. — DIMENSIONS   OP   THE   REMAINING   PARTS  OF  THE 
AIR-PUMP  MACHINERY. 

Diameter  of  air-pump  side  rods  at  ends,  in  inches  =  '039  x  D. 

Breadth  of  butt,  in  inches  ==  *046  X  D. 

Thickness  of  butt,  in  inches  =  -037  X  D. 

Mean  thickness  of  strap  at  cutter,  in  inches  =  '019  X  D. 

Mean  thickness  of  strap  below  cutter,  in  inches  =  '014  X  D. 

Depth  of  gibs  and  cutter,  in  inches  =  "048  X  D. 

Thickness  of  gibs  and  cutter  in  inches  =  D  -f-  100. 

MARINE  AND  LAND  ENGINES. — AREA  OF  STEAM  PORTS. 

Area  of  each  steam  port,  in  square  inches  =  11  X  I  X  D2  -5- 

1800  -f  8. 

MARINE  AND  LAND  ENGINES. — DIMENSIONS  OF  BRANCH  STEAM  PIPES. 


Diameter  of  each  branch  steam  pipe  =  \/  -00498  X  I  x  D2  x  10-2. 

MARINE  ENGINE. — DIMENSIONS  OF  SEVERAL  OF  THE  PIPES  CONNECTED 
WITH  THE  ENGINE. 


Diameter  of  waste  water  pipe,  in  inches  =  1-2  X  \/  H.P. 
Area  of  foot-valve  passage,  in  square  inches  =  1*8  X  H.P.  -j-  8. 
Area  of  injection  pipe,  in  square  inches  =  -069  X  H.P.  -f-  2-81. 
Diameter  of  feed  pipe,  in  inches  =  \/  -04  X  H.P.  -f  3. 
Diameter  of  waste  steam  pipe  in  inches  =\/-375x H.P. +16-875. 

MARINE  AND  LAND  ENGINES. — DIMENSIONS  OF  SAFETY-VALVES. 


Diam.  of  safety-valve,  when  one  only  is  used  =N/-5xH.P. 
Diam.  of  safety-valve,  when  two  are  used  =  v/-25x  H.P. +11*25. 
Diam.  of  safety-valve,  when  three  are  used  =  \/-167xH.P.  +  7'5. 
Diam.  of  safety-valve,  when  four  are  used  =  v/*125x H.P. +5-625. 

LAND  ENGINE. — DIMENSIONS  OF  MAIN  BEAM. 

Depth  of  web  of  main  beam  across  centre  = 

&  3  X  D2  X  half  length  of  main  beam  in  feet. 
Depth  of  main  beam  at  ends  =. 

&  -192  X  D2  X  half  length  of  main  beam,  in  feet. 

LAND  AND  MARINE  ENGINES. — CONTENT  OF  FEED-PUMP. 

Content  of  feed-pump,  in  cubic  inches  =  D2  X  I  -4-  30. 

LAND  ENGINES. — CONTENT  OF  COLD  WATER  PUMP. 

Content  of  cold  water  pump,  in  cubic  feet  =  D2  X  I  -5-  4400 


160  THE   PRACTICAL   MODEL   CALCULATOR. 

LAND  ENGINES. — DIMENSIONS  OF  CRANK. 

Thickness  of  large  eye  of  crank,  in  inches  = 


x  (1-561  x  R2  +  -1235  D2)  -i-  (R  x  666-283). 
Breadth  of  web  of  crank  at  fly-wheel  shaft  centre,  in  inches  = 

<&  D2  x  %/  (1-561  x  R2  -f  -1235  x  D2)  -r-  23-04. 
Thickness  of  web  of  crank  at  fly-wheel  shaft  centre,  in  inches  = 

^~I)2  x  V  (1-561  x  R2  +  -1235  x  D2)  +  184-32. 

LAND.  ENGINES. — DIMENSIONS  OF  FLY-WHEEL  SHAFT. 

Diameter  of  fly-wheel  shaft,  when  of  cast  iron  =  3025  X 


DIMENSIONS  OF  PARTS  6F  LOCOMOTIVES. 

DIAMETER   OF    CYLINDER. 

IN  locomotive  engines",  the  diameter  of  the  cylinder  varies  less 
than  either  the  land  or  the  marine  engine.  In  few  of  the  locomotive 
engines  at  present  in  use  is  the  diameter  of  the  cylinder  greater 
than  16  inches,  or  less  than  12  inches.  The  length  of  the  stroke  of 
nearly  all  the  locomotive  engines  at  present  in  use  is  18  inches,  and 
there  are  always  two  cylinders,  which  are  generally  connected  to 
cranks  upon  the  axle,  standing  at  right  angles  with  one  another. 

AREA  OF  INDUCTION  PORTS. 

RULE. — To  find  the  size  of  the  steam  ports  for  the  locomotive 
engine. — Multiply  the  square  of  the  diameter  of  the  cylinder  by 
•068.  The  product  is  the  proper  size  of  the  steam  ports  in  square 
inches. 

Required  the  proper  size  of  the  steam  ports  of  a  locomotive 
engine  whose  diameter  is  15  inches.  Here,  according  to  the  rule, 
size  of  steam  ports  =  -068  X  15  X  15  =  -068  X  225  =  15-3  square 
inches,  or  between  15J  and  15J  square  inches. 

After  having  determined  the  area  of  the  ports,  we  may  easily 
find  the  depth  when  the  length  is  given,  oi»,  conversely,  the  length 
when  the  depth  fs  given.  Thus,  .suppose  we  knew  that  the  length 
was  8  inches,  then  we  find  that  the  depth  should  be  15*3  -r-  8  = 
1-9125  inches,  or  nearly  2  inches;  or  suppose  we  knew  the  depth 
was  2  inches,  then  we  would' find  that  the  length  was  15-3  -r-  2  = 
7*65  inches,  or  nearly  7f  inches. 

AREA  OF  EDUCTION  PORTS. 

The  proper  area  for  the  eduction  ports  may  be  found  from  the  fol- 
lowing rule. 

RULE. —  To  find  the  area  of  the  eduction  ports. — Multiply  the 
square  of  the  diameter  of  the  cylinder  in  inches  by  -128.  The 
product  is  the  area  of  the  eduction  ports  in  square  inches. 

Required  the  area  of  the  eduction  ports  of  a  locomotive  engine, 


THE    STEAM    ENGINE.  161 

when  the  diameter  of  the  cylinders  is  13  inches.     In  this  example 
we  have,  according  to  the  rule, 

Area  of  eduction  port  =  -128  x  132  =  -128  X  169  =  21-632 
inches,  or  between  21J  and  21f  square  inches. 

BREADTH  OP  BRIDGE  BETWEEN  PORTS. 

The  breadth  of  the  bridges  between  the  eduction  port  and  the 
induction  ports  is  usually  between  J  inch  and  1  inch. 

DIAMETER  OF  BOILER. 

\ 

It  is  obvious  that  the  diameter  of  the  boiler  may  vary  very  con- 
siderably ;  but  it  is  limited  chiefly  by  considerations  of  strength ; 
and  3  feet  are  found  a  convenient  diameter.  Rules  for  the  strength 
of  boilers  will  be  given  hereafter. 

RULE. —  To  find  the  inside  diameter  of  the  boiler. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  3-11.  The  product  is  the 
inside  diameter  of  the  boiler  in  inches. 

Required  the  inside  diameter  of  the  boiler  for  a  locomotive 
engine,  the  diameter  of  the  cylinders  being  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

Inside  diameter  of  boiler  =  15  X  3*11  =  46-65  inches, 
or  about  3  feet  lOf  inches. 

LENGTH  OP  BOILER. 

The  length  of  the  boiler  is  usually  in  practice  between  8  feet  and 
8}  feet. 

DIAMETER  OF  STEAM  DOME,  INSIDE. 

It  is  obvious  that  the  diameter  of  the  steam  dome  may  be  varied 
considerably,  according  to  circumstances  ;  but  the  first  indication 
is  to  make  it  large  enough.  It  is  usual,  however,  in  practice,  to 
proportion  the  diameter  of  the  steam  dome  to  the  diameter  of  the 
cylinder;  and  there  appears  to  be  no  great  objection  to  this.  The 
following  rule  will  be  found  to  give  the  diameter  of  the  dome 
usually  adopted  in  practice. 

RULE. —  To  find  the  diameter  of  the  steam  dome. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  1-43.  The  product  is  the 
diameter  of  the  dome  in  inches. 

Required  the  diameter,  of  the  steam  dome  for  a  locomotive  engine 
whose  diameter  of  cylinders  is  13  inches.  In  this  example  we 
have,  according  to  the  rule, 

Diameter  of  steam  dome  =  1-43  x  13  =  18-59  inches, 
or  about  18J  inches. 

HEIGHT  OF  STEAM  DOME. 

The  height  of  the  steam  dome  may  vary.  Judging  from  prac- 
tice, it  appears  that  a  uniform  height  of  2J  feet  would  answer 
very  well. 

o2  11 


162  THE    PRACTICAL   MODEL    CALCULATOR. 

DIAMETER   OF    SAFETY-VALVE. 

In  practice  the  diameter  of  the  safety-valve  varies  considerably. 
The  following  rule  gives  the  diameter  of  the  safety-valve  usually 
adopted  in  practice. 

RULE. —  To  find  the  diameter  of  the  safety-valve. — Divide  the 
diameter  of  the  cylinder  in  inches  by  4.  The  quotient  is  the  dia- 
meter of  the  safety-valve  in  inches. 

Required  the  diameter  of  the  safety-valves  for  the  boiler  of  a 
locomotive  engine,  the  diameter  of  the  cylinder  being  13  inches. 
Here,  according  to  the  rule,  diameter  of  safety-valve  =  13  -r-  4  =  3J 
inches.  A  larger  size,  however,  is  preferable,  as  being  less  likely 
to  stick. 

DIAMETER   OF   VALVE    SPINDLE. 

The  following  rule  will  be  found  to  give  the  correct  diameter  of 
the  valve  spindle.  It  is  entirely  founded  on  practice. 

RULE. — To  find  the  diameter  of  the  valve  spindle. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  -076.  The  product  is  the 
proper  diameter  of  the  valve  spindle. 

Required  the  diameter  of  the  valve  spindle  for  a  locomotive 
engine  whose  cylinders'  diameters  are  13  inches. 

In  this  example  we  have,  according  to  the  rule,  diameter  of  valve 
spindle  =  13  X  -076  =  *988  inches,  or  very  nearly  1  inch. 

DIAMETER   OF   CHIMNEY. 

It  is  usual  in  practice  to  make  the  diameter  of  the  chimney  equal 
to  the  diameter  of  the  cylinder.  Thus  a  locomotive  engine  whose 
cylinders'  diameters  are  15  inches  would  have  the  inside  diameter 
of  the  chimney  also  15  inches,  or  thereabouts.  This  rule  has,  at 
least,  the  merit  of  simplicity. 

AREA   OF   FIRE-GRATE. 

The  following  rule  determines  the  area  of  the  fire-grate  usually 
given  in  practice.  We  may  remark,  that  the  area  of  the  fire-grate 
in  practice  follows  a  more  certain  rule  than  any  other  part  of  the 
engine  appears  to  do  ;  but  it  is  in  all  cases  much  too  small,  arid 
occasions  a  great  loss  of  power  by  the  urging  of  the  blast  it  renders 
necessary,  and  a  rapid  deterioration  of  the  furnace  plates  from 
excessive  heat.  There  is  no  good  reason  why  the  furnace  should 
not  be  nearly  as  long  as  the  boiler :  it  would  then  resemble  the 
furnace  of  a  marine  boiler,  and  be  as  manageable. 

RULE. — To  find  the  area  of  the  fire-grate. — Multiply  the  diameter 
of  the  cylinder  in  inches  by  *77.  ^he  product  is  the  area  of  the  fire- 
grate in  superficial  feet. 

Required  the  area  of  the  fire-grate  of  a  locomotive  engine,  the 
diameters  of  the  cylinders  being  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

Area  of  fire-grate  =  '77  X  15  =  11-55  square  feet, 
or  about  11  j  square  feet.     Though  this  rule,  however,  represents 


THE   STEAM   ENGINE.  163 

the  usual  practice,  the  area  of  the  fire-grate  should  not  be  contingent 
upon  the  size  of  the  cylinder,  but  upon  the  quantity  of  steam  to  be 
raised. 


AREA  OF  HEATING  SURFACE. 


In  the  construction  of  a  locomotive  engine,  one  great  object  is  to 
obtain  a  boiler  which  will  produce  a  sufficient  quantity  of  steam  with 
as  little  bulk  and  weight  as  possible.  This  object  is  admirably  ac- 
complished in  the  construction  of  the  boiler  of  the  locomotive  en- 
gine. This  little  barrel  of  tubes  generates  more  steam  in  an  hour 
than  was  formerly  raised  from  a  boiler  and  fire  occupying  a  con- 
siderable house.  This  favourable  result  is  obtained  simply  by  ex- 
posing the  water  to  a  greater  amount  of  heating  surface. 

In  the  usual  construction  of  the  locomotive  boiler,  it  is  obvious 
that  we  can  only  consider  four  of  the  six  faces  of  the  inside  fire-box 
as  effective  heating  surface ;  viz.  the  crown  of  the  box,  and  the 
three  perpendicular  sides.  The  circumferences  of  the  tubes  are  also 
effective  heating  surface ;  so  that  the  whole  effective  heating  sur- 
face of  a  locomotive  boiler  may  be  considered  to  be  the  four  faces 
of  the  inside  fire-box,  plus  the  sura  of  the  surfaces  of  the  tubes. 
Understanding  this  to  be  the  effective  heating  surface,  the  following 
rule  determines  the  average  amount  of  heating  surface  usually  given 
in  practice. 

RULE. — To  find  t7ie  effective  heating  surface. — Multiply  the  square 
of  the  diameter  of  the  cylinder  in  inches  by  5 ;  divide  the  product 
by  2.  The  quotient  is  the  area  of  the  effective  heating  surface  in 
square  feet. 

Required  the  effective  heating  surface  of  the  boiler  of  a  locomotive 
engine,  the  diameters  of  the  cylinders  being  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

Effective  heating  surface  =  152  X  5  H-  2  =  225  X  5  -*-  2  =  1125  -f- 
2  =  562J  square  feet. 

According  to  the  rule  which  we  have  given  for  the  fire-grate,  the 
area,  of  the  fire-grate  for  this  boiler  would  be  about  11 J  square  feet. 
We  may  suppose,  therefore,  the  area  of  the  crown  of  the  box  to  be 
12  square  feet.  The  area  of  the  three  perpendicular  sides  of  the 
inside  fire-box  is  usually  three  times  the  area  of  the  crown ;  so  that 
the  effective  heating  surface  of  the  fire-box  is  48  square  feet.  Hence 
the  heating  surface  of  the  tubes  =  526*5  —  48  =  478*5  square  feet. 
The  inside  diameters  of  the  tubes  are  generally  about  If  inches ; 
and  therefore  the  circumference  of  a  section  of  these  tubes,  ac- 
cording to  the  table,  is  5*4978  inches.  Hence,  supposing  the 
tube  to  be  8J  feet  long,  the  surface  of  one  —  5*4978  X  8J  H-  12  == 
•45815  X  8J  =  3*8943  square  feet ;  and,  therefore,  the  number  of 
tubes  —  478*5  -r-3-8943  =  123  nearly.  The  amount  of  heating  sur- 
face, however,  like  that  of  grate  surface,  is  properly  a  function  of 
the  quantity  of  steam  to  be  raised,  and  the  proportions  of  both, 
given  hereafter,  will  be  found  to  answer  well  for  boilers  of  every 
description. 


164  THE    PRACTICAL   MODEL    CALCULATOR. 

AREA   OP   WATER-LEVEL. 

This,  of  course,  varies  with  the  different  circumstances  of  the 
boiler.  The  average  area  may  be  found  from  the  following  rule. 

RULE. — To  find  the  area  of  the  water-level. — Multiply  the  diame- 
ter of  the  cylinder  in  inches  by  2*08.  The  product  is  the  area  of 
the  water-level  in  square  feet. 

Required  the  area  of  the  water-level  for  a  locomotive  engine, 
whose  cylinders'  diameters  are  14  inches. 

In  this  case  we  have,  according  to  the  rule, 

Area  of  water-level  =  14  X  2-08  =  29-12  square  feet. 

CUBICAL    CONTENT    OF    WATER   IN    BOILER. 

This,  of  course,  varies  not  only  in  different  boilers,  but  also  in 
the  same  boiler  at  different  times.  The  following  rule  is  supposed 
to  give  the  average  quantity  of  water  in  the  boiler. 

RULE. — To  find  the  cubical  content  of  the  water  in  the  boiler. — 
Multiply  the  square  of  the  diameter  of  the  cylinder  in  inches  by  9 : 
divide  the  product  by  40.  The  quotient  is  the  cubical  content  of 
the  water  in  the  boiler  in  cubic  feet. 

Required  the  average  cubical  content  of  the  water  in  the  boiler 
of  a  locomotive  engine,  the  diameters  of  the  cylinders  being  14 
inches.  In  this  example  we  have,  according  to  the  rule, 

Cubical  content  of  water  =  9  X  142  -f-  40  =  44-1  cubic/ feet. 

CONTENT   OF   FEED-PUMP. 

In  the  locomotive  engine,  the  feed-pump  is  generally  attached  to 
the  cross-head,  and  consequently  it  has  the  same  stroke  as  the  pis- 
ton. As  we  have  mentioned  before,  the  stroke  of  the  locomotive 
engine  is  generally  in  practice  18  inches.  Hence,  assuming  the 
stroke  of  the  feed-pump  to  be  constantly  18  inches,  it  only  remains 
for  us  to  determine  the  diameter  of  the  ram.  It  may  be  found  from 
the  following  rule. 

RULE. — To  find  the  diameter  of  the  feed-pump  ram. — Multiply 
the  square  of  the  diameter  of  the  cylinder  in  inches  by  -Oil.  The 
product  is  the  diameter  of  the  ram  in  inches. 

Required  the  diameter  of  the  ram  for  the  feed-pump  for  a  loco- 
motive engine  whose  diameter  of  cylinder  is  14  inches.  In  this 
example  we  have,  according  to  the  rule, 

Diameter  of  ram  =  -Oil  x  142  =  -Oil  x  196  =  2-156  inches, 
or  between  2  and  2J  inches. 

CUBICAL   CONTENT    OF    STEAM   ROOM. 

The  quantity  of  steam  in  the  boiler  varies  not  only  for  different 
boilers,  but  even  for  the  same  boiler  in  different  circumstances. 
But  when  the  locomotive  is  in  motion,  there  is  usually  a  certain 
proportion  of  the  boiler  filled  with  the  steam.  Including  the  dome 
and  the  steam  pipe,  the  content  of  the  steam  room  will  be  found 
usually  to  be  somewhat  less  than  the  cubical  content  of  the  water. 


THE   STEAM   ENGINE.  165 

But  as  it  is  desirable  that  it  should  be  increased,  we  give  the  fol- 
lowing rule. 

RULE. — 'To  find  the  cubical  content  of  the  steam  room. — Multiply 
the  square  of  the  diameter  of  the  cylinder  in  inches  by  9 ;  divide 
the  product  by  40.  The  quotient  is  the  cubical  content  of  the 
steam  room  in  cubic  feet. 

Required  the  cubical  content  of  the  steam  room  in  a  locomotive 
boiler,  the  diameters  of  the  cylinders  being  12  inches. 

In  this  example  we  have,  according  to  the  rule, 

Cubical  content  of  steam  room  =  9  X  12*  -T-  40  =  9  X  144  -r-  40  = 
32*4  cubic  feet. 

CUBICAL   CONTENT   OF   INSIDE   FIRE-BOX   ABOVE   FIRE-BARS. 

The  following  rule  determines  the  cubical  content  of  fire-box 
usually  given  in  practice. 

RULE. — To  find  the  cubical  content  of  inside  fire-box  above  fire- 
bars.— Divide  the  square  of  the  diameter  of  the  cylinder  in  inches 
by  4.  The  quotient  is  the  content  of  the  inside  fire-box  above  fire- 
bars in  cubic  feet. 

Required  the  content  of  inside  fire-box  above  fire-bars  in  a  loco- 
motive engine,  when  the  diameters  of  the  cylinders  are  each  15 
inches. 

In  this  example  we  have,  according  to  the  rule, 

Content  of  inside  fire-box  above  fire-bars  =  152  -4-4  =  225  -s-  4  = 
56J  cubic  feet. 

THICKNESS    OF   THE    PLATES   OF   BOILER. 

In  general,  the  thickness  of  the  plates  of  the  locomotive  boiler  is 
|  inch.  In  some  cases,  however,  the  thickness  is  only  &  inch. 

INSIDE   DIAMETER   OF    STEAM  PIPE. 

The  diameter  usually  given  to  the  steam  pipe  of  the  locomotive 
engine  may  be  found  from  the  following  rule. 

RULE. —  To  find  the  diameter  of  the  steam  pipe  of  the  locomotive 
engine. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  -03.  The  product  is  the  diameter  of  the  steam  pipe  in 
inches. 

Required  the  diameter  of  the  steam  pipe  of  a  locomotive  engine, 
the  diameter  of  the  cylinder  being  13  inches.  Here,  according  to 
the  rule,  diameter  of  steam  pipe  =  -03  X  132*=  -03  X  169  =  5-07 
inches ;  or  a  very  little  more  than  5  inches.  The  steam  pipe  is 
usually  made  too  small  in  engines  intended  for  high  speeds. 

DIAMETER   OF   BRANCH   STEAM   PIPES. 

The  following  rule  gives  the  usual  diameter  of  the  branch  steam 
pipe  for  locomotive  engines. 

RULE. — To  find  the  diameter  of  the  branch  steam  pipe  for  the  lo- 
comotive engine. — Multiply  the  square  of  the  diameter  of  the  cylin- 
der in  inches  by  -021.  The  product  is  the  diameter  of  the  branch 
steam  pipe  for  the  locomotive  engine  in  inches. 


166  THE  PRACTICAL  MODEL  CALCULATOR. 

Required  the  diameter  of  the  branch  steam  pipes  for  a  locomo- 
tive engine,  when  the  cylinder's  diameter  is  15  inches.  Here,  ac- 
cording to  the  rule,  diameter  of  branch  pipe  =  -021  X  152  =  -021  x 
225  =  4-725  inches,  or  about  4f  inches. 

DIAMETER   OF   TOP   OF  BLAST   PIPE. 

The  diameter  of  the  top  of  the  blast  pipe  may  be  found  from  the 
following  rule. 

RULE. —  To  find  the  diameter  of  the  top  of  the  blast  pipe. — Mul- 
tiply the  square  of  the  diameter  of  the  cylinder  in  inches  by  0-17. 
The  product  is  the  diameter  of  the  top  of  the  blast  pipe  in  inches. 

The  diameter  of  a  locomotive  engine  is  13  inches ;  required  the 
diameter  of  the  blast  pipe  at  top.  Here,  according  to  the  rule, 
diameter  of  blast  pipe  at  top  =  -017  X  132  =  -017  X  169  =2-873 
inches,  or  between  2  j  and  3  inches ;  but  the  orifice  of  the  blast 
pipe  should  always  be  made  as  large  as  the  demands  of  the  blast 
will  permit. 

DIAMETER   OF   FEED   PIPES. 

There  appear  to  be  no  theoretical  considerations  which  would 
lead  us  to  determine  exactly  the  proper  size  of  the  feed  pipes. 
Judging  from  practice,  however,  the  following  rule  will  be  found  to 
give  the  proper  dimensions. 

RULE. —  To  find  the  diameter  of  the  feed  pipes. — Multiply  the 
diameter  of  the  cylinder  in  inches  by  *141.  The  product  is  the 
proper  diameter  of  the  feed  pipes. 

'Required  the  diameter  of  the  feed  pipes  for  a  locomotive  engine, 
the  diameter  of  the  cylinder  being  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

Diameter  of  feed-pipe  =  15  X  -141  =  2-115  inches, 
or  between  2  and  2J  inches. 

DIAMETER   OF   PISTON   ROD. 

The  diameter  of  the  piston  rod  for  the  locomotive  engine  is 
usually  about  one-seventh  the  diameter  of  the  cylinder.  Making 
practice  our  guide,  therefore,  we  have  the  following  rule. 

RULE. —  To  find  the  diameter  of  the  piston  rod  for  the  locomotive 
engine. — Divide  the  diameter  of  the  cylinder  in  inches  by  7.  The 
quotient  is  the  diameler  of  the  piston  rod  in  inches. 

The  diameter  of  the  cylinder  of  a  locomotive  engine  is  15  inches ; 
required  the  diameter  of  the  piston  rod.  Here,  according  to  the 
rule,  diameter  of  piston  rod  =15  -5-  7  =  2}  inches. 

THICKNESS   OF   PISTON. 

The  thickness  of  the  piston  in  locomotive  engines  is  usually  about 
two-sevenths  of  the  diameter  of  the  cylinder.  Making  practice  our 
guide,  therefore,  we  have  the  following  rule. 

RULE. —  To  find  the  thickness  of  the  piston  in  the  locomotive  en- 
gine.— Multiply  the  diameter  of  the  cylinder  in  inches  by  2  ;  divide 


THE    STEAM    ENGINE.  167 

the  product  by  7.     The  quotient  is  the  thickness  of  the  piston  in 
inches. 

The  diameter  of  the  cylinder  of  a  locomotive  engine  is  14  inches ; 
required  the  thickness  of  the  piston.  Here,  according  to  the  rule, 
thickness  of  piston  =  2xl4-f-7=4  inches. 

DIAMETER   OF   CONNECTING   RODS   AT    MIDDLE. 

The  following  rule  gives  the  diameter  of  the  connecting  rod  at 
middle.  The  rule,  we  may  remark,  is  entirely  founded  on  practice. 

RULE. — To  find  the  diameter  of  the  connecting  rod  at  middle  of 
the  locomotive  engine. — Multiply  the  diameter  of  the  cylinder  in 
inches  by  -21.  The  product  is  the  diameter  of  the  connecting  rod 
at  middle  in  inches. 

Required  the  diameter  of  the  connecting  rods  at  middle  for  a 
locomotive  engine,  the  diameter  of  the  cylinders  being  twelve 
inches. 

For  this  example  we  have,  according  to  the  rule, 

Diameter  of  connecting  rods  at  middle  =  12  X  -21  =  2-52  inches, 
or  2J  inches. 

DIAMETER   OF  BALL   ON   CROSS-HEAD    SPINDLE.       . 

The  diameter  of  the  ball  on  the  cross-head  spindle  may  be  found 
from  the  following  rule. 

RULE. — To  find  the  diameter  of  the  ballon  cross-head  spindle  of 
a  locomotive  engine. — Multiply  the  diameter  of  the  cylinder  in 
inches  by  -23.  The  product  is  the  diameter  of  the  ball  on  the 
cross-head  spindle. 

Required  the  diameter  of  the  ball  on  the  cross-head  spindle  of  a 
locomotive  engine,  when  the  diameter  of  the  cylinder  is  15  inches. 
Here,  according  to  the  rule, 

Diameter  of  ball  =  -23  X  15  =  345  inches,  or  nearly  3J  inches. 

DIAMETER   OF   THE   INSIDE   BEARINGS   OF   THE    CRANK   AXLE. 

It  is  obvious  that  the  inside  bearings  of  the  crank  axle  of  the 
locomotive  engine  correspond  to  the  paddle-shaft  journal  of  the 
marine  engine,  and  to  the  fly-wheel  shaft  journal  of  the  land-engine. 
We  may  conclude,  therefore,  that  the  proper  diameter  of  these  bear- 
ings ought  to  depend  jointly  upon  the  length  of  the  stroke  and  the 
diameter  of  the  cylinder.  In  the  locomotive  engine  the  stroke  is 
usually  18  inches,  so  that  we  may  consider  that  the  diameter  of  the 
bearing  depends  solely  upon  the  diameter  of  the  cylinder.  The 
following  rule  will  give  the  diameter  of  the  inside  bearing. 

RULE. — To  find  the  diameter  of  the  inside  bearing  for  the  loco- 
motive engine. — Extract  the  cube  root  of  the  square  of  the  diameter 
of  the  cylinder  in  inches ;  multiply  the  result  by  -96.  The  product 
is  the  proper  diameter  of  the  inside  bearing  of  the  crank  axle  for  the 
locomotive  engine. 

Required  the  diameter  of  the  inside  bearing  of  the  crank  axle 


168  THE  PRACTICAL  MODEL  CALCULATOR. 

for  a  locomotive  engine  whose  cylinders  are  of  13-inch  diameters. 
In  this  example  we  have,  according  to  the  rule, 

13  =  diameter  of  cylinder  in  inches. 

2? 

169  =  square  of  the  diameter  of  cylinder. 

0  0  169(5-5289  =  ^169 

5  25  125_ 

5  25  44000 

_5  50  41375 

10  7500  2625 

5  775  1820 

150  8275  805 

_5  800  726 

155  9075  79 

_5  _3 

160  910 

5  _3 

165  913 

and  diameter  of  bearing  =  5-5289  X  -96  =  5*31  inches  nearly;  or 
between  5J  and  5J  inches. 

DIAMETER   OF   THE   OUTSIDE    BEARINGS   OP   THE   CRANK  AXLE. 

The  crank  axle,  in  addition  to  resting  upon  the  inside  bearings, 
is  sometimes  also  made  to  rest  partly  upon  outside  bearings. 
These  outside  bearings  are  added  only  for  the  sake  of  steadiness, 
and  they  do  not  need  to  be  so  strong  as  the  inside  bearings.  The 
proper  size  of  the  diameter  of  these  bearings  may  be  found  from 
the  following  rule. 

RULE. —  To  find  the  diameter  of  outside  bearings  for  the  locomo- 
tive engine. — Multiply  the  square  of  the  diameters  of  the  cylinders 
in  inches  by  *396  ;  extract  the  cube  root  of  the  product.  The  result 
is  the  diameter  of  the  outside  bearings  in  inches. 

Required  the  proper  diameter  of  the  outside  bearings  for  a  loco- 
motive engine,  the  diameter  of  its  cylinders  being  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

15  =  diameter  of  cylinders  in  inches. 
J5 

225  =  square  of  diameter  of  cylinder. 
•396  =  constant  multiplier. 

89-1 


THE   STEAM   ENGINE.  169 


0       0       89-1(4466  =  ^89-1 
£      16       64 

4  16  25100 

4  32  21184 

8  4800  3916 

4  4%  3528 

120  "5296  "388 

_4  512  358 

124  5808 

_4  8 

128  588 

_i     _J? 
132     596 

Hence  diameter  of  outside  bearing  =  4466  inches,  or  very 
nearly  4J  inches. 

DIAMETER   OP  PLAIN   PART   OP  CRANK   AXLE. 

It  is  usual  to  make  the  plain  part  of  crank  axle  of  the  same  sec- 
tional area  as  the  inside  bearings.  Hence,  to  determine  the  sec- 
tional area  of  the  plain  part  when  it  is  cylindrical,  we  have  the  fol- 
lowing rule. 

RULE. — To  determine  the  diameter  of  the  plain  part  of  crank  axle 
for  the  locomotive  engine. — Extract  the  cube  root  of  the  square  of 
the  diameter  of  the  cylinder  in  inches;  multiply  the  result  by  -96. 
The  product  is  the  proper  diameter  of  the  plain  part  of  the  crank 
axle  of  the  locomotive  engine  in  inches. 

Required  the  diameter  of  the  plain  part  of  the  crank  axle  for  the 
locomotive  engine,  whose  cylinders'  diameters  are  14  inches.  In 
this  example  we  have,  according  to  the  rule, 

14  =  diameter  of  cylinder  in  inches. 
JL4 

196  =  square  of  the  diameter  of  cylinder. 

0  -0  196(5-808 

5  25  125 

5  25  71-000 

_5  50  70-112 

10  7500  -888 

5  1264 

150       8764 
__8       1328 

158      10092 

_8 

166 

_8 

174 


170          THE  PRACTICAL  MODEL  CALCULATOR. 

Hence  the  plain  part  of  crank  axle  =  5-808  X  -96  =  5-58  nearly, 
or  a  little  more  than  5J  inches. 

DIAMETER   OF   CRANK   PIN. 

The  following  rule  gives  the  proper  diameter  of  the  crank  pin.  It  is 
obvious  that  the  crank  pin  of  the  locomotive  engine  is  not  altogether 
analogous  to  the  crank  pin  of  the  marine  or  land  engine,  and,  like 
them,  ought  to  depend  upon  the  diameter  of  the  cylinder,  as  it  is 
usually  formed  out  of  the  solid  axle. 

RULE. — To  find  the  diameter  of  the  crank  pin  for  the  locomotive 
engine. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -404. 
The  product  is  the  diametor  of  the  crank  pin  in  inches. 

Required  the  diameter  of  the  crank  pin  of  a  locomotive  engine 
whose  cylinders'  diameters  are  15  inches. 

In  this  example  we  have,  according  to  the  rule, 

Diameter  of  crank  pin  =  15  X  -404  =  6-06  inches,  or  about  6 
inches. 

LENGTH   OF   CRANK   PIN. 

The  length  of  the  crank  pin  usually  given  in  practice  may  be 
found  from  the  following  rule. 

RULE. — To  find  the  length  of  the  crank  pin. — Multiply  the  di- 
ameter of  the  cylinder  in  inches  by  -233.  The  product  is  the 
length  of  the  crank  pins  in  inches. 

Required  the  length  of  the  crank  pins  for  a  locomotive  engine 
with  a  diameter  of  cylinder  of  13  inches. 

In  this  example  we  have,  according  to  the  rule, 

Length  of  crank  pin  =  13  X  -233  =  3-029  inches, 
or  about  3  inches.     The  part  of  the  crank  axle  answering  to  the 
crank  pin  is  usually  rounded  very  much  at  the  corners,  both  to  give 
additional  strength,  and  to  prevent  side  play. 

These  then  are  the  chief  dimensions  of  locomotive  engines  ac- 
cording to  the  practice  most  generally  followed.  The  establish- 
ment of  express  trains  and  the  general  exigencies  of  steam  locomo- 
tion are  daily  introducing  innovations,  the  eifect  of  which  is  to  make 
the  engines  of  greater  size  and  power :  but  it  cannot  be  said  that  a 
plan  of  locomotive  engine  has  yet  been  contrived  that  is  free  from 
grave  objections.  The  most  material  of  these  defects  is  the  neces- 
sity that  yet  exists  of  expending  a  large  proportion  of  the  power  in 
the  production  of  a  draft ;  and  this  evil  is  traceable  to  the  inade- 
quate area  of  the  fire-grate,  which  makes  an  enormous  rush  of  air 
through  the  fire  necessary  to  accomplish  the  combustion  of  the  fuel 
requisite  for  the  production  of  the  steam.  To  gain  a  sufficient  area 
of  fire-grate,  an  entirely  new  arrangement  of  engine  must  be 
adopted :  the  furnace  must  be  greatly  lengthened,  and  perhaps  it 
may  be  found  that  short  upright  tubes,  or  the  very  ingenious  ar- 
rangement of  Mr.  Dimpfell,  of  Philadelphia,  may  be  introduced 
with  advantage.  Upright  tubes  have  been  found  to  be  more 
effectual  in  raising  steam  than  horizontal  tubes ;  but  the  tube 
plate  in  the  case  of  upright  tubes  would  be  more  liable  to  burn. 


THE    STEAM    ENGINE.  171 

We  here  give  the  preceding  rules  in.  formulas,  in  the  belief  that 
those  well  acquainted  with  algebraic  symbols  prefer  to  have  a  rule 
expressed  as  a  formula,  as  they  can  thus  see  at  once  the  different 
operations  to  be  performed.  In  the  following  formulas  we  denote 
the  diameter  of  the  cylinder  in  inches  by  D. 

LOCOMOTIVE  ENGINE. — PARTS  OF  THE  CYLINDER. 

Area  of  induction  ports,  in  square  inches  =  *068  X  D2. 
Area  of  eduction  ports,  in  square  inches  =  *128  X  D2. 
Breadth  of  bridge  between  ports  between  f  inch  and  1  inch. 

LOCOMOTIVE  ENGINE. — PARTS  OF  BOILER. 

Diameter  of  boiler,  in  inches  =  3-11  X  D. 
Length  of  boiler  between  8  feet  and  12  feet. 
Diameter  of  steam  dome,  inside,  in  inches  =  1*43  X  D. 
Height  of  steam  dome  =  2J  feet. 
Diameter  of  safety  valve,  in  inches  =  D  -4-  4. 
Diameter  of  valve  spindle,  in  inches  =  *076  X  D. 
Diameter  of  chimney,  in  inches  =  D. 
Area  of  fire-grate,  in  square  feet  =  *77  X  D. 
Area  of  heating  surface,  in  square  feet  =  5  X  D2  -4-  2. 
Area  of  water  level,  in  square  feet  =  2-08  X  D. 
Cubical  content  of  water  in  boiler,  in  cubic  feet  =  9  X  D2  -4-  40. 
Diameter  of  feed-pump  ram,  in  inches  =  -Oil  X  D2. 
Cubical  content  of  steam  room,  in  cubic  feet  =  9  X  D2  -4-  40. 
Cubical  content  of  inside  fire-box  above  fire  bars,  in  cubic  feet  = 
D2  -4-4. 

Thickness  of  the  plates  of  boiler  =  f  inch. 

LOCOMOTIVE  ENGINE. — DIMENSIONS  OF  SEVERAL  PIPES. 

Inside  diameter  of  steam  pipe,  in  inches  =  *03  X  D2. 
Inside  diameter  of  branch  steam  pipe,  in  inches  =  *021  X  D2. 
Inside  diameter  of  the  top  of  blast  pipe  =  *017  X  D2. 
Inside  diameter  of  the  feed  pipes  =  -141  x  D. ' 

LOCOMOTIVE  ENGINE. — DIMENSIONS  OF  SEVERAL  MOVING  PARTS. 

Diameter  of  piston  rod,  in  inches  =  D  -4-  7. 

Thickness  of  piston,  in  inches  =  2  D  -4-  7. 

Diameter  of  connecting  rods  at  middle,  in  inches  =  *21  X  D. 

Diameter  of  the  ball  on  cross-head  spindle,  in  inches  =  *23  X  D. 

Diameter  of  the  inside  bearings  of  the  crank  axle,  in  inches  = 
•96  x  y  D2. 

Diameter  of  the  plain  part  of  crank  axle,  in  inches  =  *96  X  \^  D2. 

Diameter  of  the  outside  bearings  of  the  crank  axle,  in  inches  = 
V  -396  x  D2. 

Diameter  of  crank  pin,  in  inches  =  -404  X  D. 

Length  of  crank  pin,  in  inches  =  -233  x  D. 


172 


THE  PRACTICAL  MODEL  CALCULATOR. 


TABLE  of  the  Pressure  of  Steam,  in  Inches  of  Mercury ',  at  dif- 
ferent Temperatures. 


Tempe- 
rature, 
Fahren- 
heit. 

Daltoa. 

Ure. 

Young. 

Ivory. 

Tredgold. 

Southern. 

Robison. 

Watt. 

0° 

0-08 

... 

... 

... 

... 

... 

... 

10 

0-12 

... 

... 

... 

... 

... 

... 

20 

0-17 

... 

0-11 

... 

... 

... 

... 

32 

0-26 

0-20 

0-18 

... 

0-17 

0-16 

0-00 

... 

40 

0-34 

0-25 

0-20 

... 

0-24 

0-22 

0-10 

... 

50 

0-49 

0-36 

0-36 

0-36 

0-37 

0-33 

0-20 

... 

60 

0-65 

0-52 

0-53 

... 

0-55 

0-48 

0-35 

70 

0-87 

0-73 

0-75 

0-73 

0-78 

0-68 

0-55 

0-77 

80 

1-16 

1-01 

1-05 

... 

1-11 

0-95 

0-82 

... 

90 

1-59 

1-36 

1-44 

1-36 

1-53 

1-34 

1-18 

... 

100 

2-12 

1-86 

1-95 

... 

2-08 

1-84 

1-60 

1-55 

110 

2-79 

2-45 

2-62 

2-46 

2-79 

2-56 

2-25 

... 

120 

3-63 

3-30 

3-46 

... 

3-68 

3-46 

3-00 

... 

130 

4-71 

4-37 

4-54 

4-41 

4-81 

4-43 

3-95 

140 

6-05 

5-78 

5-88 

... 

6-21 

5-75 

5-15 

5-14 

150 

7-73 

7-53 

7-55 

7-42 

7-94 

7-46 

6-72 

... 

160 

9-79 

9-60 

9-62 

... 

10-05 

9-52 

8-65 

8-92 

170 

12-31 

12-05 

12-14 

12-05 

12-60 

12-14 

11-05 

11-37 

180 

15-38 

15-16 

15-23 

... 

15-67 

15-20 

14-05 

12-73 

190 

18-98 

19-00 

18-96 

18-93 

19-00 

... 

17-85 

19-00 

200 

23-51 

23-60 

23-44 

23-71 

22-65 

210 

28-82 

28-88 

'28-81 

28-81 

28-86 

... 

28-62 

212 

30-00 

30-00 

30-00 

30-00 

30-00 

30-00 

30-00 

29-40 

220 

35-18 

35-54 

35-19 

... 

34-92 

... 

35-8 

33-65 

230 

44-60 

43-10 

42-47 

*42-63 

42-00 

44-5 

40 

240 

53-45 

51-70 

51-66 

50-24 

... 

54-9 

49-0 

TABLE  of  the  Temperature  of  Steam  at  different  Pressures  in  At 
mospheres. 


Pressure  in 
Atmospheres. 

French 
Academy. 

Dr.  Ure. 

Young. 

Ivory. 

Tredgold. 

Southern. 

Robison. 

Watt. 

Franklin 
Institute. 

1st  At. 
'2d   At. 
3d  At. 
4th  At. 
5th  At. 
6th  At. 
7th  At. 
8th  At. 
9th  At. 
10th  At 

212-0° 
250-5 
275-2 
293-7 
308-8 
320-4 
331-7 
342-0 
350-0 
358-9 

212° 

250-0 
275-0 
291-5 
304-5 
315-5 
325-5 
336-0 
345-0 

212° 
240-3 
271 
288 
302 

212° 
249 

290 
337 

212° 
250 

274 
294 
309 
322 

342*' 

250-3 
293-4 

343-6 

212° 
267 

212° 
252-5 

212° 
250-0 
275-2 
291-5 
304-5 
315-5 
326-5 
336-0 
345-0 
352-5 

llth  At 

366-8 

1'Hh  At 

374-0 

372 

13th  At 

380-6 

14th  At 

386-9 

15th  At 

392-8 

383-8 

16th  At 

398-5 

17th  At 

403-8 

18th  At 

408-9 

19th  At 

413-9 

20th  At. 
30th  At 

418-5 
457-2 

... 

... 

... 

414 

... 

... 

405 

40th  At 

466-6 

50th  At 

510-6 

i 

THE   STEAM   ENGINE. 


173 


TABLE  of  the  Expansion  of  Air  by  Heat. 


Fahren. 
32 

1000 

Fahren. 

61 

1069 

Fahren. 
90 

1132 

33  ... 

1002 

62 

1071 

.91  . 

1134 

34  

1004 

63  

1073 

92  .  . 

1136 

35 

1007 

64 

1075 

93 

1138 

36  

1009 

65   

1077 

94 

1140 

37  

1012 

66  

1030 

95 

1142 

38 

1015 

67  . 

1080 

96 

1144 

39  

1018 

68  

1034 

97  

1146 

40  

1021 

69  

1087 

98 

1148 

41  

1023 

70  

1089 

99  

1150 

42 

1025 

71  

1091 

100 

1152 

43  

1027 

72  

1093 

110  . 

1173 

44  

1030 

73  

1095 

120  

1194 

45 

1032 

74  .  . 

1097 

130 

1215 

46  

1034 

75  

1099 

140 

1235 

47  

1036 

76  

1101 

150  

1255 

48  

1038 

77  

1104 

160  

1275 

49  

1040 

78 

1106 

170 

1295 

50  

1043 

79 

1108 

180 

1315 

51  

1045 

80  

1110 

190  ...,  

1334 

52 

1047 

81 

1112 

200 

1364 

53  .... 

1050 

82 

1114 

210 

1372 

54  

1052 

83 

1116 

212 

1376 

55  

1055 

84  .  . 

1118 

302  

1558 

56  

1057 

85  

1121 

392  

1739 

57  

1059 

86  

1123 

482  

1919 

58  

1062 

87 

1125 

572 

2098 

59  

1064 

88 

1128 

680  

2312 

60  

1066 

89  

1130 

STRENGTH   OF    MATERIALS. 


The  chief  materials,  of  which  it  is  necessary  to  record  the  strength 
in  this  place,  are  cast  and  malleable  iron ;  and  many  experiments 
have  been  made  at  different  times  upon  each  of  these  substances, 
though  not  with  any  very  close  correspondence.  The  following  is 
a  summary  of  them : — 


Materials. 

c 

s 

E 

M 

f  from 

16300  1 

Iron,  cast  \  *™n 

36000  / 

8100 

69120000 

5530000 

Malleable   .  .      

60000 

9000 

91440000 

6770000 

Wire  

80000 

The  first  column  of  figures,  marked  C,  contains  the  mean  strength 
of  cohesion  on  an  inch  section  of  the  material ;  the  second,  marked 
S,  the  constant  for  transverse  strains ;  the  third,  marked  E,  the 
constant  for  deflections ;  and  the  fourth,  marked  M,  the  modulus 
of  elasticity.  The  introduction  of  the  liot  blast  iron  brought  with 
it  the  impression  that  it  was  less  strong  than  that  previously  in  use, 
and  the  experiments  which  had  previously  been  confided  in  as 
giving  results  near  enough  the  truth,  for  all  practical  purposes, 
were  no  longer  considered  to  be  applicable  to  the  new  state  of 
things.  New  experiments  were  therefore  made.  The  following 
Table  gives,  we  have  no  doubt,  results  as  nearly  correct  as  can  be 
required  or  attained : — 


174 


THE   PRACTICAL   MODEL   CALCULATOR. 


RESULTS   OP    EXPERIMENTS   ON    THE   STRENGTH    AND   OTHER   PRO- 
PERTIES  OF  CAST   IRON. 

IN  the  following.  Table  each  bar  is  reduced  to  exactly  one  inch 
square ;  and  the  transverse  strength,  which  may  be  taken  as  a 
criterion  of  the  value  of  each  Iron,  is  obtained  from  a  mean  between 
the  experiments  upon  it; — first  on  bars  4  ft.  6  in.  between  the 
supports ;  and  next  on  those  of  half  the  length,  or  2  ft.  3  in.  be- 
tween the  supports.  All  the  other  results  are  deduced  from  the 
4  ft.  6  in.  bars.  In  all  cases  the  weights  were  laid  on  the  middle 
of  the  bar. 


NAME  or  ISOH. 

1 

Modulus  of  elasticity  in 
Ibs.  per  sq.  inch,  or  stiff- 

Breaking  weight  in  Ibs.  of 
bars  4  ft.  6  in.  between 
supports. 

•ss^ 

III 

t'i 

fii 

Mean  breaking  weight  in 
(SO 

Ultimate  deflection  of  4  ft. 
6  in.  bars,  in  parts  of  an 

Power  of  the  4  ft.  6  in. 
bars  to  resist  impact. 

Colour. 

TV  k             '      N            lr    N    T 

7-030 
7-122 
7-251 
7-300 
7-056 
7-069 
7-8 
7-066 
7-071 
7-049 
7-108 
7-055 
7-079 
7-017 
7-017 
7-059 
7-038 
7-038 
7-113 
7-080 
7-159 
7-285 
7-017 
7-031 
7-028 
7-094 
7-087 
7-038 
7-007 
7-080 
6-979 
7-051 
6-998 
7-080 
6-975 
7-031 
7-041 
7-038 
6-928 
7-007 
7-12S 
7-069 
6-953 
7-185  '. 
6-9C9 
6-955 
6-916 
6-957 
6-976 
6-916 

18470000 
17211000 
22473650 
2-2733400 
17873100 
16802000 
15379500 
15163000 
16490000 
•14607000 
16301000 
14509500 
15381200 
14911666 
14852000 
14307500 
15193000 
13959500 
14003550 
13815500 
142S1466 
22907700 
13894000 
13112666 
15787666 
16246966 
16534000 
13971500 
13845866 
13136500 
15394766 
15S52500 
13730500 
15452500 
15280900 
15241000 
14953333 
14211000 
12586500 
15012000 
15510066 
17036000 
132!  14400 
161  56133 
14322500 
14304000 
12259500 
11539333 
11974500 
13341633 

510 
567 
537 
543 
520 
505 
500 
495 
483 
441 
478 
462 
463 
466 
457 
453 
438 
453 
443 
441 
433 
448 
427 
460 
444 
444 
456 
440 
430 
439 
432 
427 
436 
401 
408 
419 
413 
408 
446 
422 
464 
430 
417 
404 
409 
408 
402 
392 
353 
378 

532 
595 

517 
534 
529 
515 
487 
495 
529 
470 
483 

453 
455 
457 
473 
455 
464 
457 
4(54 

467 
434 

443 
430 
444 
454 
441 
449 
449 

403 
455 
439 
446 
446 
408 
430 
385 
408 
419 
432 
424 
418 
404 

386 
387 

600 
581 

537 
530 
527 
517 
502 
491 
489 
485 
474 
472 
463 
459 
45< 
455 
455 
454 
453 
449 
448 
448 
447 
447 
444 
443 
443 
442 
442 
440 
440 
438 
436 
432 
431 
429 
429 
427 
427 
426 
424 
419 
418 
418 
416 
413 
403 
392 
369 
357 

1-530 

1-747 
1-09 

1-005 
1-365 
1-599 
1-815 
1-764 
1-581 
1-621 
1-512 
1-852 
1-55 
1-748 
1-730 
1-811 
1484 
1-957 
1-734 
1-759 
1-726 
•790 
1-557 
1-825 
1-414 
1-336 
1-469 
1-887 
1.687 
1-857 
1-443 
1-368 
1-64 
1-516 
1-251 
1-358 
1-339 
1-512 
2-224 
1-450 
1-532 
1-231 
1-570 
1-222 
1-882 
1-470 
1-762 
1-890 
1-525 
1-366 

991 
992 
589 
549 
710 
807 
889 
872 
765 
718 
729 
855 
721 
815 
791 
822 
650 
886 
770 
777 
747 
353 
998 
841 
629 
593 
674 
830 
727 
816 
625 
585 
721 
699 
511 
570 
554 
618 
992 
621 
716 
530 
656 
494 
771 
600 
709 
742 
538 
517 

Gray 

Whitish  gray 
White 
White 
Whitish  gray 
Dullish  gray 
Dark  gray 
Bluish  gray 
Dark  gray 
Gray 
Dull  gray    • 
Dark  gray 
Gray 
Light  gray 
Light  gray 
Dark  gray 
Bluish  gray 
Dark  gray 
Bright  gray 
Light  gray 
Bright  gray 
Light  gray 
Light  gray 
Light  gray 
Dark  gray 
Gray 
Dull  gray 
Bluish  gray 
Gray 
Dull  blue 
Light  gray 
Gray 
Dull  gray 
Light  gray 
Light  gray 
Dull  gray 
Light  gray 
Bluish  gray 
Gray 
Gray 
Whitish  gray 
G  raj- 
Bluish  gray 
Dark  gray 
Bright  gray 
Gray 
Bluish  gray 
Bluish  gray 
Gray 
Light  gray 

Ponkey,  No.  3.  Cold  Blast  
Devon,  No.  3.  Hot  Blast*  

Pattison,  N.  J.  Hot  Blast*  

Bute,  No.  1.  Cold  Blast  
Wind  Mill  End,  No.  2.  Cold  Blast 
Old  Park,  No.  2.  Cold  Blast 
Beaufort,  No.  2.  Hot  Blast  
Low  Moor,  No.  2.  Cold  Blast  
Buffery,  No.  1.  Cold  Blast*   
Brimbo  \o  i;)  Cold  Blast  

Apedale,  No.  2.  Hot  Blast  
Oldberry,  No.  2.  Cold  Blast  •  •  -  - 

Maesteg,  No.  2  
Muirkirk,  No.  1.  Cold  Blast*-  •  •  • 
Adelphi,  No.  2.  Cold  Blast  
Blania  No  3  Cold  Blast  

Gartsherrie,  No.  3.  Hot  Blast  • 

Lane  End  No  2   

Carron,  No.  3.  Cold  Blast*  
Dundyvan,  No.  3.  Cold  Blast-  •  •  • 
Alaesteg  (Marked  Red)  

Corbyns  Hall,  No  2  

Milton  No  3  Hot  Blast  

Buffery,  No.  1.  Hot  Blast*  
Level,  No.  1.  Hot  Blast  

W  S  S    No  2  

Eagle  Foundry.  No.  2.  Hot  Blast 
Elsicar  No  2  Cold  Blast  

Varteg,  No.  2.  Hot  Blast  •'• 
Coltham,  No.  1.  Hot  Blast  
Carroll,  No.  2.  Cold  Blast  
Muirkirk,  No.  1.  Hot  Blast*-  •  • 

Coed-Talon,'  No.  2.  Hot  Blast*  • 
Coed-Talon,  No.  2.  Cold  Blast*- 
Monkland,  No.  2.  Hot  Blast    •  • 
Ley's  Works,  No.  1.  Hot  Blast- 
Milton,  No.  1.  Hot  Blast  
Plaskynaston,  No.  2.  Hot  Blast  • 

The  irons  with  asterisks  are  taken  from  Experiments  on  Hot  and 
Cold  Blast  Iron. 


THE    STEAM    ENGINE. 


175 


RULE. — To  find  from  the  above  Table  the  breaking  weight  in 
rectangular  bars,  generally.  Calling  b  and  d  the  breadth  and 
depth  in  inches,  and  I  the  distance  between  the  supports,  in  feet, 

4-5  X  b  d2  S 
and  putting  4*5  for  4  ft.  6  in.,  we  have  -      — j —     -  =  breaking 

weight  in  Ibs., — the  value  of  S  being  taken  from  the  above  Table. 
For  example: — What  weight  would  be  necessary  to  break  a  bar 
of  Low  Moor  Iron,  2  inches  broad,  3  inches  deep,  and  6  feet  be- 
tween the  supports  ?  According  to  the  rule  given  above,  we  have 
b  =  2  inches,  d  =  3  inches,  I  =  6  feet,  S  =  472  from  the  Table. 

4-5  x  b  d2  S      4-5  x  2  x  32  x  472      ^n 
Then —     -  =  -  — —  -  =  6372  Ibs.,  the  break- 


l 


6 


ing  weight. 

TABLE  of  the  Cohesive  Power  of  Bodies  whose  Gross  Sectional  Areas 
equal  one  Square  Inch. 


METALS. 


Swedish  bar  iron , 

Russian      do 

English       do , 

Cast  steel 

Blistered  do 

Shear  do 

Wrought  copper 

Hard  gun-metal 

Cast  copper 

Yellow  brass,  cast 

Cast  iron 

Tin,  cast 

Bismuth,  cast 

Lead,  cast 

Elastic  power  or  direct  tension  of  wrought  iron, 
medium  quality 


Cohesive  Power 
in  Ibs. 


65,000 

59,470 

56,000 

134,256 

133,152 

127,632 

33,892 

36,368 

19,072 

17,968 

17,628 

4,736 

3,250 

1,824 

22,400 


NOTE. — A  bar  of  iron  is  extended  -000096,  or  nearly  one  ten- 
thousandth  part  of  its  length,  for  every  ton  of  direct  strain  per 
square  inch  of  sectional  area. 


CENTRE   OF  GRAVITY. 


The  centre  of  gravity  of  a  body  is  that  point  within  it  which 
continually  endeavours  to  gain  the  lowest  possible  situation  ;  or  it  is 
that  point  on  which  the  body,  being  freely  suspended,  will  remain 
at  rest  in  all  positions.  The  centre  of  gravity  of  a  body  does  not 
always  exist  within  the  matter  of  which  the  body  is  composed, 
there  being  bodies  of  such  forms  as  to  preclude  the  possibility  of 
this  being  the  case,  but  it  must  either  be  surrounded  by  the  con- 
stituent matter,  or  so  placed  that  the  particles  shall  be  symmetri- 
cally situated,  with  respect  to  a  vertical  line  in  which  the  position 
of  the  centre  occurs.  Thus,  the  centre  of  gravity  of  a  ring  is  not 
in  the  substance  of  the  ring  itself,  but,  if  the  ring  be  uniform,  it  will 
be  in  the  axis  of  its  circumscribing  cylinder  ;  and  if  the  ring  varies 


176          THE  PRACTICAL  MODEL  CALCULATOR. 

in  form  or  density,  it  will  be  situated  nearest  to  those  parts  where 
the  weight  or  density  is  greatest.  Varying  the  position  of  a  body 
will  not  cause  any  change  in  the  situation  of  the  centre  of  gravity : 
for  any  change  of  position  the  body  undergoes  will  only  have  the 
effect  of  altering  the  directions  of  the  sustaining  forces,  which  will 
still  preserve  their  parallelism.  When  a  body  is  suspended  by  any 
other  point  than  its  centre  of  gravity,  it  will  not  rest  unless  that 
centre  be  in  the  same  vertical  line  with  the  point  of  suspension ; 
for,  in  every  other  position,  the  force  which  is  intended  to  insure 
the  equilibrium  will  not  directly  oppose  the  resultant  of  gravity 
upon  the  particles  of  the  body,  and  of  course  the  equilibrium  will 
not  obtain ;  the  directions  of  the  forces  of  gravity  upon  the  con- 
stituent particles  are  all  parallel  to  one  another  and  perpendicular 
to  the  horizon.  If  a  heavy  body  be  sustained  by  two  or  more 
forces,  their  lines  of  direction  must  meet  either  at  the  centre  of 
gravity,  or  in  the  vertical  line  in  which  it  occurs. 

A  body  cannot  descend  or  fall  downwards,  unless  it  be  in  such 
a  position  that  by  its  motion  the  centre  of  gravity  descends.  If  a 
body  stands  on  a  plane,  and  a  line  be  drawn  perpendicular  to  the 
horizon,  and  if  this  perpendicular  line  fall  within  the  base  of  the 
body,  it  will  be  supported  without  falling ;  but  if  the  perpendicular 
falls  without  the  base  of  the  body,  it  will  overset.  For  when  the 
perpendicular  falls  within  the  base,  the  body  cannot  be  moved  at  all 
without  raising  the  centre  of  gravity ;  but  when  the  perpendicular 
falls  without  the  base  towards  any  side,  if  the  body  be  moved 
towards  that  side,  the  centre  of  gravity  will  descend,  and  conse- 
quently the  body  will  overset  in  that  direction.  If  a  perpendicular 
to  the  horizon  from  the  centre  of  gravity  fall  upon  the  extremity 
of  the  base,  the  body  may  continue  to  stand,  but  the  least  force 
that  can  be  applied  will  cause  it  to  overset  in  that  direction  ;  and 
the  nearer  the  perpendicular  is  to  any  side  the  easier  the  body  will 
be  made  to  fall  on  that  side,  but  the  nearer  the  perpendicular  is  to 
the  middle  of  the  base  the  firmer  the  body  will  stand.  If  the 
centre  of  gravity  of  a  body  be  supported,  the  whole  body  is  sup- 
ported, and  the  place  of  the  centre  of  gravity  must  be  considered 
as  the  place  of  the  body,  and  it  is  always  in  a  line  which  is  perpen- 
dicular to  the  horizon. 

In  any  two  bodies,  the  common  centre  of  gravity  divides  the 
line  that  joins  their  individual  centres  into  two  parts  that  are  to 
one  another  reciprocally  as  the  magnitudes  of  the  bodies.  The 
products  of  the  bodies  multiplied  by  their  respective  distances  from 
the  common  centre  of  gravity  are  equal.  If  a  weight  be  laid 
upon  any  point  of  an  inflexible  lever  which  is  supported  at  the 
ends,  the  pressure  on  each  point  of  the  support  will  be  inversely 
as  the  respective  distances  from  the  point  where  the  weight  is 
applied.  In  a  system  of  three  bodies,  if  a  line  be  drawn  from  the 
centre  of  gravity  of  any  one  of  them  to  the  common  centre  of  the 
other  two,  then  the  common  centre  of  all  the  three  bodies  divides 
the  line  into  two  parts  that  are  to  each  other  reciprocally  as  the 


THE    STEAM    ENGINE.  177 

magnitude  of  the  body  from  which  the  line  is  drawn  to  the  sum  of 
the  magnitudes  of  the  other  two ;  and,  consequently,  the  single 
body  multiplied  by  its  distance  from  the  common  centre  of  gravity 
is  equal  to  the  sum  of  the  other  bodies  multiplied  by  the  distance 
of  their  common  centre  from  the  common  centre  of  the  system. 

If  there  be  taken  any  point  in  the  straight  line  or  lever  joining 
the  centres  of  gravity  of  two  bodies,  the  sum  of  the  two  products 
of  each  body  multiplied  by  its  distance  from  that  point  is  equal  to 
the  product  of  the  sum  of  the  bodies  multiplied  by  the  distance  of 
their  common  centre  of  gravity  from  the  same  point.  The  two 
bodies  have,  therefore,  the  same  tendency  to  turn  the  lever  about 
the  assumed  point,  as  if  they  were  both  placed  in  their  common 
centre  of  gravity.  Or,  if  the  line  with  the  bodies  moves  about  the 
assumed  point,  the  sum  of  the  momenta  is  equal  to  the  momentum 
of  the  sum  of  the  bodies  placed  at  their  common  centre  of  gravity. 
The  same  property  holds  with  respect  to  any  number  of  bodies 
whatever,  and  also  when  the  bodies  are  not  placed  in  the  line,  but 
iu  perpendiculars  to  it  passing  through  the  bodies.  If  any  plane 
pass  through  the  assumed  point,  perpendicular  to  the  line  in  which 
it  subsists,  then  the  distance  of  the  common  centre  of  gravity  of 
all  the  bodies  from  that  plain  is  equal  to  the  sum  of  all  the 
momenta  divided  by  the  sum  of  all  the  bodies.  We  may  here 
specify  the  positions  of  the  centre  of  gravity  in  several  figures  of 
very  frequent  occurrence. 

In  a,  straight  line,  or  in  a  straight  bar  or  rod  of  uniform  figure 
and  density,  the  position  of  the  centre  of  gravity  is  at  the  middle 
of  its  length.  In  the  plane  of  a  triangle  the  centre  of  gravity  is 
situated  in  the  straight  line  drawn  from  any  one  of  the  angles  to 
the  middle  of  the  opposite  side,  and  at  two-thirds  of  this  line  dis- 
tant from  the  angle  where  it  originates,  or  one-third  distant  from 
the  base.  In  the  surface  of  a  trapezium  the  centre  of  gravity  is  in 
the  intersections  of  the  straight  lines  that  join  the  centres  of  the 
opposite  triangles  made  by  the  two  diagonals.  The  centre  of 
gravity  of  the  surface  of  a  parallelogram  is  at  the  intersection  of 
the  diagonals,  or  at  the  intersection  of  the  two  lines  which  bisect 
the  figure  from  its  opposite  sides.  In  any  regular  polygon  the 
centre  of  gravity  is  at  the  same  point  as  the  centre  of  magnitude. 
In  a  circular  arc  the  position  of  the  centre  of  gravity  is  distant 
from  the  centre  of  the  circle  by  the  measure  of  a  fourth  propor- 
tional to  the  arc,  radius,  and  chord.  In  a  semicircular  arc  the 
position  of  the  centre  of  gravity  is  distant  from  the  centre  by  the 
measure  of  a  third  proportional  to  the  arc  of  the  quadrant  and  the 
radius.  In  the  sector  of  a  circle  the  position  of  the  centre  of 
gravity  is  distant  from  the  centre  of  the  circle  by  a  fourth  propor- 
tional to  three  times  the  arc  of  the  sector,  the  chord  of  the  arc, 
and  the  diameter  of  the  circle.  In  a  circular  segment,  the  position 
of  the  centre  of  gravity  is  distant  from  the  centre  of  the  circle  by 
a  space  which  is  equal  to  the  cube  or  third  power  of  the  chord 
divided  by  twelve  times  the  area  of  the  segment.  In  a  semicircle 

12 


178          THE  PRACTICAL  MODEL  CALCULATOR. 

the  position  of  the  centre  of  gravity  is  distant  from  the  centre  of 
the  circle  by  a  space  which  is  equal  to  four  times  the  radius  divided 
by  the  constant  number  8-1416  X  3  =  9-4248.  In  a  parabola  the 
position  of  the  centre  of  gravity  is  distant  from  the  vertex  by 
three-fifths  of  the  axis.  In  a  semi-parabola  the  position  of  the 
centre  of  gravity  is  at  the  intersection  of  the  co-ordinates,  one  of 
which  is  parallel  to  the  base,  and  distant  from  it  by  two-fifths  of 
the  axis,  and  the'  other  parallel  to  the  axis,  but  distant  from  it  by 
three-eighths  of  the  semi-base. 

The  centres  of  gravity  of  the  surface  of  a  cylinder,  a  cone,  and 
conic  frustum,  are  respectively  at  the  same  distances  from  the  origin 
as  are  the  centres  of  gravity  of  the  parallelogram,  the  triangle,  and 
the  trapezoid,  which  are  sections  passing  along  the  axes  of  the  re- 
spective solids.  The  centre  of  gravity  of  the  surface  of  a  spheric  seg- 
ment is  at  the  middle  of  the  versed  sine  or  height.  The  centre  of 
gravity  of  the  convex  surface  of  a  spherical  zone  is  at  the  middle  of 
that  portion  of  the  axis  of  the  sphere  intercepted  by  its  two  bases. 
In  prisms  and  cylinders  the  position  of  the  centre  of  gravity  is  at  the 
middle  of  the  straight  line  that  joins  the  centres  of  gravity  of  their 
opposite  ends.  In  pyramids  and  cones  the  centre  of  gravity  is  in 
the  straight  line  that  joins  the  vertex  with  the  centre  of  gravity 
of  the  base,  and  at  three-fourths  of  its  length  from  the  vertex,  and 
one-fourth  from  the  base.  In  a  sernisphere,  or  semispheroid,  the 
position  of  the  centre  of  gravity  is  distant  from  the  centre  by  three- 
eighths  of  the  radius.  In  a  parabolic  conoid  the  position  of  the 
centre  of  gravity  is  distant  from  the  base  by  one-third  of  the  axis, 
or  two-thirds  of  the  axis  distant  from  the  vertex.  There  are 
several  other  bodies  and  figures  of  which  the  position  of  the  centre 
of  gravity  is  known  ;  but  as  the  position  in  those  cases  cannot  be 
defined  without  algebra,  we  omit  them. 

CENTRIPETAL  AND  CENTRIFUGAL  FORCES. 

Central  forces  are  of  two  kinds,  centripetal  and  centrifugal. 
Centripetal  force  is  that  force  by  which  a  body  is  attracted  or 
impelled  towards  a  certain  fixed  point  as  a  centre,  and  that  point 
towards  which  the  body  is  urged  is  called  the  centre  of  attraction 
or  the  centre  of  force.  Centrifugal  force  is  that  force  by  which  a 
body  endeavours  to  recede  from  the  centre  of  attraction,  and  from 
which  it  would  actually  fly  off  in  the  direction  of  a  tangent  if  it 
were  not  prevented  by  the  action  of  the  centripetal  force.  These 
two  forces  are  therefore  antagonistic  ;  the  action  of  the  one  being 
directly  opposed  to  that  of  the  other.  It  is  on  the  joint  action  of 
these  two  forces  that  all  curvilinear  motion  depends.  Circular  motion 
is  that  affection  of  curvilinear  motion  where  the  body  is  constrained 
to  move  in  the  circumference  of  a  circle:  if  it  continues  to  move  so 
as  to  describe  the  entire  circle,  it  is  denominated  rotatory  motion,  and 
the  body  is  said  to  revolve  in  a  circular  orbit,  the  centre  of  which  is 
called  the  centre  of  motion.  In  all  circular  motions  the  deflection 
or  deviation  from  the  rectilinear  course  is  constantly  the  same  at 


THE   STEAM   ENGINE.  179 

every  point  of  the  orbit,  in  which  case  the  centripetal  and  centri- 
fugal forces  are  equal  to  one  another.  In  circular  orbits  the  cen- 
tripetal forces,  by  which  equal  bodies  placed  at  equal  distances 
from  the  centres  of  force  are  attracted  or  drawn  towards  those 
centres,  are  proportional  to  the  quantities  of  matter  in  the  central 
bodies.  This  is  manifest,  for  since  all  attraction  takes  place 
towards  some  particular  body,  every  particle  in  the  attracting  body 
must  produce  its  individual  effect ;  consequently,  a  body  containing 
twice  the  quantity  of  matter  will  exert  twice  the  attractive  energy, 
and  a  body  containing  thrice  the  quantity  of  matter  will  operate 
with  thrice  the  attractive  force,  and  so  on  according  to  the  quantity 
of  matter  in  the  attracting  body. 

Any  body,  whether  large  or  small,  when  placed  a^the  same  dis- 
tance from  the  centre  of  force,  is  attracted  or  drawn  through  equal 
spaces  in  the  same  time  by  the  action  of  the  central  body.  This 
is  obvious  from  the  consideration  that  although  a  body  two  or  three 
times  greater  is  urged  with  two  or  three  times  greater  an  attractive 
force,  yet  there  is  two  or  three  times  the  quantity  of  matter  to  be 
moved  ;  and,  as  we  have  shown  elsewhere,  the  velocity  generated 
in  a  given  time  is  directly  proportional  to  the  force  by  which  it  is 
generated,  and  inversely  as  the  quantity  of  matter  in  the  moving 
or  attracted  body.  But  the  force  which  in  the  present  instance  is 
the  weight  of  the  body  is  proportional  to  the  quantity  of  matter 
which  it  contains ;  consequently,  the  velocity  generated  is  directly 
and  inversely  proportional  to  the  quantity  of  matter  in  the 
attracted  body,  and  is,  therefore,  a  given  or  a  constant  quantity. 
Hence,  the  centripetal  force,  or  force  towards  the  centre  of  the 
circular  orbit,  is  not  measured  by  the  magnitude  of  the  revolving 
body,  but  only  by  the  space  which  it  describes  or  passes  over  in  a 
given  time.  When  a  body  revolves  in  a  circular  orbit,  and  is 
retained  in  it  by  means  of  a  centripetal  force  directed  to  the 
centre,  the  actual  velocity  of  the  revolving  body  at  every  point  of 
its  revolution  is  equal  to  that  which  it  would  acquire  by  falling 
perpendicularly  with  the  same  uniform  force  through  one-fourth  of 
the  diameter,  or  one-half  the  radius  of  its  orbit ;  and  this  velocity 
is  the  same  as  would  be  acquired  by  a  second  body  in  falling 
through  half  the  radius,  whilst  the  first  body,  in  revolving  in  its 
orbit,  describes  a  portion  of  the  circumference  which  is  equal  in 
length  to  half  the  diameter  of  the  circle.  Consequently,  if  a  body 
revolves  uniformly  in  the  circumference  of  a  circle  by  means  of  a 
given  centripetal  force,  the  portion  of  the  circumference  which  it 
describes  in  any  time  is  a  mean  proportional  between  the  diameter 
of  the  circle  and  the  space  which  the  body  would  descend  perpen- 
dicularly in  the  same  time,  and  with  the  same  given  force  continued 
uniformly. 

The  periodic  time,  in  the  doctrine  of  central  forces,  is  the  time 
occupied  by  a  body  in  performing  a  complete  revolution  round  the 
centre,  when  that  body  is  constrained  to  move  in  the  circumference 
by  means  of  a  centripetal  force  directed  to  that  point ;  and  when 


180          THE  PRACTICAL  MODEL  CALCULATOR. 

the  body  revolves  in  a  circular  orbit,  the  periodic  time,  or  the 
time  of  performing  a  complete  revolution,  is  expressed  by  the  term 

n  t  %/  ^,  and  the  velocity  or  space  passed  over  in  the  time  f  will  be 

5 

v/  ds ;  in  which  expressions  d  denotes  the  diameter  of  the  circular 
orbit  described  by  the  revolving  body,  s  the  space  descended  in  any 
time  by  a  body  falling  perpendicularly  downwards  with  the  same 
uniform  force,  t  the  time  of  descending  through  the  space,  s  and  ?t 
the  circumference  of  a  circle  whose  diameter  is  unity.  If  several 
bodies  revolving  in  circles  round  the  same  or  different  centres  be 
retained  in  their  orbits  by  the  action  of  centripetal  forces  directed 
to  those  points,  the  periodic  times  will  be  directly  as  the  square 
roots  of  the  rajjii  or  distances  of  the  revolving  bodies,  and  inversely 
as  the  square  roots  of  the  centripetal  forces,  or,  what  is  the  same 
thing,  the  squares  of  the  periodic  times  are  directly  as  the  radii, 
and  inversely  as  the  centripetal  forces. 

CENTRE    OF    GYRATION. 

The  centre  of  gyration  is  that  point  in  which,  if  all  the  consti- 
tuent particles,  or  all  the  matter  contained  in  a  revolving  body,  or 
system  of  bodies,  were  concentrated,  the  same  angular  velocity 
would  be  generated  in  the  same  time  by  a  given  force  aeting  at  any 
place  as  would  be  generated  by  the  same  force  acting  similarly  on 
the  body  or  system  itself  according  to  its  formation. 

The  angular  motion  of  a  body,  or  system  of  bodies,  is  the  motion 
of  a  line  connecting  any  point  with  the  centre  or  axis  of  motion, 
and  is  the  same  in  all  parts  of  the  same  revolving  system. 

In  different  unconnected  bodies,  each  revolving  about  a  centre, 
the  angular  velocity  is  directly  proportional  to  the  absolute  velo- 
city, and  inversely  as  the  distance  from  the  centre  of  motion  ;  so 
that,  if  the  absolute  velocities  of  the  revolving  bodies  be  propor- 
tional to  their  radii  or  distances,  the  angular  velocities  will  be 
equal.  If  the  axis  of  motion  passes  through  the  centre  of  gravity, 
then  is  this  centre  called  the  principal  centre  of  gyration. 

The  distance  of  the  centre  of  gyration  from  the  point  of  suspen- 
sion, or  the  axis  of  motion  in  any  body  or  system  of  bodies,  is  a 
geometrical  mean  between  the  centres  of  gravity  and  oscillation 
from  the  same  point  or  axis  ;  consequently,  having  found  the  dis- 
tances of  these  centres  in  any  proposed  case,  the  square  root  of 
their  product  will  give  the  distance  of  the  centre  of  gyration.  If 
any  part  of  a  system  be  conceived  to  be  collected  in  the  centre  of 
gyration  of  that  particular  part,  the  centre  of  gyration  of  the 
whole  system  will  continue  the  same  as  before ;  for  the  same  force  that 
moved  this  part  of  the  system  before  along  with  the  rest  will  move 
it  now  without  any  change  ;  and  consequently,  if  each  part  of  the 
system  be  collected  into  its  own  particular  centre,  the  common 
centre  of  the  whole  system  will  continue  the  same.  If  a  circle  be 
described  about  the  centre  of  gravity  of  any  system,  and  the  axis 
of  rotation  be  made  to  pass  through  any  point  of  the  circumference, 


THE    STEAM   ENGINE.  181 

the  distance  of  the  centre  of  gyration  from  that  point  will  always 
be  the  same. 

If  the  periphery  of  a  circle  revolve  about  an  axis  passing  through 
the  centre,  and  at  right  angles  to  its  plane,  it  is  the  same  thing  as 
if  all  the  matter  were  collected  into  any  one  point  in  the  peri- 
phery. And  moreover,  the  plane  of  a  circle  or  a  disk  containing 
twice  the  quantity  of  matter  as  the  said  periphery,  and  having  the 
same  diameter,  will  in  an  equal  time  acquire  the  same  angular 
velocity.  If  the  matter  of  a  revolving  body  were  actually  to  be 
placed  in  the  centre  of  gyration,  it  ought  either  to  be  arranged  in 
the  circumference,  or  in  two  points  of  the  circumference  diametri- 
cally opposite  to  each  other,  and  equally  distant  from  the  centre 
of  motion,  for  by  this  means  the  centre  of  motion  will  coincide 
with  the  centre  of  gravity,  and  the  body  will  revolve  without  any 
lateral  force  on  any  side.  These  are  the  chief  properties  con- 
nected with  the  centre  of  gyration,  and  the  following  are  a  few  of 
the  cases  in  which  its  position  has  been  ascertained. 

In  a  right  line,  or  a  cylinder  of  very  small  diameter  revolving 
about  one  of  its  extremities,  the  distance  of  the  centre  of  gyration 
from  the  centre  of  motion  is  equal  to  the  length  of  the  revolving 
line  or  cylinder  multiplied  by  the  square  root  of  J-.  In  the  plane 
of  a  circle,  or  a  cylinder  revolving  about  the  axis,  it  is  equal  to  the 
radius  multiplied  by  the  square  root  of  J.  In  the  circumference 
of  a  circle  revolving  about  the  diameter  it  is  equal  to  the  radius 
multiplied  by  the  square  root  of  -J.  In  the  plane  of  a  circle 
revolving  about  the  diameter  it  is  equal  to  one-half  the  radius.  In 
a  thin  circular  ring  revolving  about  one  of  its  diameters  as  an  axis 
it  is  equal  to  the  radius  multiplied  by  the  square  root  of  J.  In  a 
solid  globe  revolving  about  the  diameter  it  is  equal  to  the  radius 
multiplied  by  the  square  root  of  f.  In  the  surface  of  a  sphere 
revolving  about  the  diameter  it  is  equal  to  the  radius  multiplied  by 
the  square  root  of  f .  In  a  right  cone  revolving  about  the  axis  it 
is  equal  to  the  radius  of  the  base  multiplied  by  the  square  root  of  1%. 
In  all  these  cases  the  distance  is  estimated  from  the  centre  of 
the  axis  of  motion.  We  shall  have  occasion  to  illustrate  these  prin- 
ciples when  we  come  to  treat  of  fly-wheels  in  the  construction  of 
the  different  parts  of  steam  engines. 

When  bodies  revolving  in  the  circumferences  of  different  circles 
are  retained  in  their  orbits  by  centripetal  forces  directed  to  the 
centres,  the  periodic  times  of  revolution  are  directly  proportional 
to  the  distances  or  radii  of  the  circles,  and  inversely  as  the  veloci- 
ties of  motion ;  and  the  periodic  times,  under  like  circumstances, 
are  directly  as  the  velocities  of  motion,  and  inversely  as  the  cen- 
tripetal forces.  If  the  times  of  revolution  are  equal,  the  velocities 
and  centripetal  forces  are  directly  as  the  distances  or  radii  of  the 
circles.  If  the  centripetal  forces  are  equal,  the  squares  of  the 
times  of  revolution  and  the  squares  of  the  velocities  are  as  the  dis- 
tances or  radii  of  the  circles.  If  the  times  of  revolution  are  as 
Q 


182  THE    PRACTICAL   MODEL   CALCULATOR. 

the  radii  of  the  circles,  the  velocities  will  be  equal,  and  the  cen- 
tripetal forces  reciprocally  as  the  radii. 

If  several  bodies  revolve  in  circular  orbits  round  the  same  or 
different  centres,  the  velocities  are  directly  as  the  distances  or 
radii,  and  inversely  as  the  times  of  revolution. .  The  velocities  are 
directly  as  the  centripetal  forces  and  the  times  of  revolution.  The 
squares  of  the  velocities  are  proportional  to  the  centripetal  forces, 
and  the  distances  or  radii  of  the  circles.  When  the  velocities  are 
equal,  the  times  of  revolution  are  proportional  to  the  radii  of  the 
circles  in  which  the  bodies  revolve,  and  the  radii  of  the  circles  are 
inversely  as  the  centripetal  forces.  If  the  velocities  be  propor- 
tional to  the  distances  or  radii  of  the  circles,  the  centripetal  forces 
will  be  in  the  same  ratio,  and  the  times  of  revolution  will  be  equal. 

If  several  bodies  revolve  in  circular  orbits  about  the  same  or 
different  centres,  the  centripetal  forces  are  proportional  to  the  dis- 
tances or  radii  of  the  circles  directly,  and  inversely  as  the  squares 
of  the  times  of  revolution.  The  centripetal  forces  are  directly 
proportional  to  the  velocities,  and  inversely  as  the  times  of  revolu- 
tion. The  centripetal  forces  are  directly  as  the  squares  of  the 
velocities,  and  inversely  as  the  distances  or  radii  of  the  circles. 
When  the  centripetal  forces  are  equal,  the  velocities  are  propor- 
tional to  the  times  of  revolution,  and  the  distances  as  the  squares 
of  the  times  or  as  the  squares  of  the  velocities.  When  the  central 
forces  are  proportional  to  the  distances  or  radii  of  the  circles,  the 
times  of  revolution  are  equal.  If  several  bodies  revolve  in  circular 
orbits  about  the  sa.me  or  different  centres,  the  radii  of  the  circles 
are  directly  proportional  to  the  centripetal  forces,  and  the  squares 
of  the  periodic  times.  The  distances  or  radii  of  the  circles  are 
directly  as  the  velocities  and  periodic  times.  The  distances  or 
radii  of  the  circles  are  directly  as  the  squares  of  the  velocities,  and 
reciprocally  as  the  centripetal  forces.  If  the  distances  are  equal, 
the  centripetal  forces  are  directly  as  the  squares  of  the  velocities, 
and  reciprocally  as  the  squares  of  the  times  of  revolution ;  the 
velocities  also  are  reciprocally  as  the  times  of  revolution.  The 
converse  of  these  principles  and  properties  are  equally  true ;  and 
all  that  has  been  here  stated  in  regard  to  centripetal  forces  is 
similarly  true  of  centrifugal  forces,  they  being  equal  and  contrary 
to  each  other. 

The  quantities  of  matter  in  all  attracting  bodies,  having  other 
bodies  revolving  about  them  in  circular  orbits,  are  proportional  to 
the  cubes  of  the  distances  directly,  and  to  the  squares  of  the  times 
of  revolution  reciprocally.  The  attractive  force  of  a  body  is 
directly  proportional  to  the  quantity  of  matter,  and  inversely  as 
the  square  of  the  distance.  If  the  centripetal  force  of  a  body 
revolving  in  a  circular  orbit  be  proportional  to  the  distance  from 
the  centre,  a  body  let  fall  from  the  upper  extremity  of  the  vertical 
diameter  will  reach  the  centre  in  the  same  time  that  the  revolving 
body  describes  one-fourth  part  of  the  orbit.  The  velocity  of  the 
descending  body  at  any  point  of  the  diameter  is  proportional  to 


THE    STEAM    ENGINE.  183 

the  ordinate  of  the  circle  at  that  point ;  and  the  time  of  falling 
through  any  portion  of  the  diameter  is  proportional  to  the  arc  of 
the  circumference  whose  versed  sine  is  the  space  fallen  through. 
All  the  times  of  falling  from  any  altitudes  whatever  to  the  centre 
of  the  orbit  will  be  equal  ;  for  these  times  are  equal  to  one-fourth 
of  the  periodic  times,  and  these  times,  under  the  specified  condi- 
tions, are  equal.  The  velocity  of  the  descending  body  at  the  centre 
of  the  circular  orbit  is  equal  to  the  velocity  of  the  revolving  body. 
These  are  the  chief  principles  that  we  need  consider  regarding 
the  motion  of  bodies  in  circular  orbits ;  and  from  them  we  are  led 
to  the  consideration  of  bodies  suspended  on  a  centre,  and  made  to 
revolve  in  a  circle  beneath  the  suspending  point,  so  that  when  the 
body  describes  the  circumference  of  a  circle,  the  string  or  wire  by 
which  it  is  suspended  describes  the  surface  of  a  cone.  A  body  thus 
revolving  is  called  a  conical  pendulum,  and  this  species  of  pendu- 
lum, or,  as  it  is  usually  termed,  the  governor,  is  of  great  importance 
in  mechanical  arrangements,  being  employed  to  regulate  the  move- 
ments of  steam  engines,  water-wheels,  and  other  mechanism.  As 
we  shall  have  occasion  to  show  the  construction  and  use  of  this  in- 
strument when  treating  of  the  parts  and  proportions  of  engines,  we 
need  not  do  more  at  present  than  state  the  principles  on  which  its 
action  depends.  We  must,  however,  previously  say  a  few  words 
on  the  properties  of  the  simple  pendulum,  or  that  which,  being  sus- 
pended from  a  centre,  is  made  to  vibrate  from  side  to  side  in  the 
same  vertical  plane. 

PENDULUMS. 

If  a  pendulum  vibrates  in  a  small  circular  arc,  the  time  of  per- 
forming one  vibration  is  to  the  time  occupied  by  a  heavy  bodyun 
falling  perpendicularly  through  half  the  length  of  the  pendulum  as 
the  circumference  of  a  circle  is  to  its  diameter.  All  vibrations  of 
the  same  pendulum  made  in  very  small  circular  arcs,  are  made  in 
very  nearly  the  same  time.  The  space  described  by  a  falling  body 
in  the  time  of  one  vibration  is  to  half  the  length  of  the  pendulum 
as  the  square  of  the  circumference  of  a  circle  is  to  the  square  of 
the  diameter.  The  lengths  of  two  pendulums  which  by  vibrating 
describe  similar  circular  arcs  are  to  each  other  as  the  squares  of 
the  times  of  vibration.  The  times  of  pendulums  vibrating  in  small 
circular  arcs  are  as  the  square  roots  of  the  lengths  of  the  pendulums. 
The  velocity  of  a  pendulum  at  the  lowest  point  of  its  path  is  pro- 
portional to  the  chord  of  the  arc  through  which  it  descends  to  ac- 
quire that  velocity.  Pendulums  of  the  same  length  vibrate  in  the 
same  time,  whatever  the  weights  may  be.  From  which  we  infer, 
that  all  bodies  near  the  earth's  surface,  whether  they  be  heavy  or 
light,  will  fall  through  equal  spaces  in  equal  times,  the  resistance 
of  the  air  not  being  considered. 

The  lengths  of  pendulums  vibrating  in  the  same  time  in  different 
positions  of  the  earth's  surface  are  as  the  forces  of  gravity  in  those 
positions.  The  times  wherein  pendulums  of  the  same  length  will 
vibrate  by  different  forces  of  gravity  are  inversely  as  the  square 


184  THE    PRACTICAL    MODEL    CALCULATOR. 

roots  of  the  forces.  The  lengths  of  pendulums  vibrating  in  dif- 
ferent places  are  as  the  forces  of  gravity  at  those  places  and  the 
squares  of  the  times  of  vibration.  The  times  in  which  pendulums 
of  any  length  perform  their  vibrations  are  directly  as  the  sqhare 
roots  of  their  lengths,  and  inversely  as  the  square  roots  of  the  gravi- 
tating forces.  The  forces  of  gravity  at  different  places  on  the  earth's 
surface  are  directly  as  the  lengths  of  the  pendulums,  and  inversely 
as  the  squares  of  the  times  of  vibration.  These  are  the  chief  proper- 
ties of  a  simple  pendulum  vibrating  in  a  vertical  plane,  and  the  prin- 
cipal problems  that  arise  in  connection  with  it  are  the  following,  viz.  : 

To  find  the  length  of  a  pendulum  that  shall  make  any  number 
of  vibrations  in  a  given  time  ;  and  secondly,  having  given  the  length 
of  a  pendulum,  to  find  the  number  of  vibrations  it  will  make  in  any 
time  given. — These  are  problems  of  very  easy  solution,  and  the 
rules  for  resolving  them  are  simply  as  follow : — For  the  first,  the 
rule  is,  multiply  the  square  of  the  number  of  seconds  in  the  given 
time  by  the  constant  number  39*1015,  and  divide  the  product  by 
the  square  of  the  number  of  vibrations,  for  the  length  of  the 
pendulum  in  inches.  For  the  second,  it  is,  multiply  the  square  of 
the  number  of  seconds  in  the  given  time  by  the  constant  number 
39-1398,  divide  the  product  by  the  given  length  of  the  pendulum 
in  inches,  and  extract  the  square  root  of  the  quotient  for  the  num- 
ber of  vibrations  sought.  The  number  39-1015  is  the  length  of  a 
pendulum  in  inches,  that  vibrates  seconds,  or  sixty  times  in  a  minute, 
in  the  latitude  of  Philadelphia. 

Suppose  a  pendulum  is  found  to  make  35  vibrations  in  a  minute ; 
what  is  the  distance  from  the  centre  of  suspension  to  the  centre  of 
oscillation  ? 

Here,  by  the  rule,  the  number  of  seconds  in  the  given  time  is  60 ; 
hence  we  get  60  x  60  x  39-1015  =  1407654,  which,  being  di- 
vided by  35  x  35  =  1225,  gives  140765-4  -=-  1225  =  114-9105 
inches  for  the  length  required. 

The  length  of  a  pendulum  between  the  centre  of  suspension  and 
the  centre  of  oscillation  is  64  inches ;  what  number  of  vibrations 
will  it  make  in  60  seconds  ? 

By  the  rule  we  have  60  X  60  X  39-1015  =  1407654,  which, 
being  divided  by  64,  gives  140765-4  -r-  64  =  2199-46,  and  the 
square  root  of  this  is  2199-46  =  46-9,  number  of  vibrations 
sought.  When  the  given  time  is  a  minute,  or  60  seconds,  as  in  the 
two  examples,  proposed  above,  the  product  of  'the  constant  number 
39-1015  by  the  square  of  the  time,  or  140765*4,  is  itself  a  constant 
quantity,  which,  being  kept  in  mind,  will  in  some  measure  facilitate  the 
process  of  calculation  in  all  similar  cases.  We  now  return  to  the 
consideration  of  the  conical  pendulum,  or  that  in  which  the  ball  re- 
volves about  a  vertical  axis  in  the  circumference  of  a  circular  plane 
which  is  parallel  to  the  horizon. 

§  CONICAL   PENDULUM. 

If  a  pendulum  be  suspended  from  the  upper  extremity  of  a  ver- 
tical axis,  arid  be  made  to  revolve  about  that  axis  by  a  conical  mo- 


THE    STEAM    ENGINE.  185 

tion,  which  constrains  the  revolving  body  to  move  in  the  circum- 
ference of  a  circle  whose  plane  is  parallel  to  the  horizon,  then  the 
time  in  which  the  pendulum  performs  a  revolution  about  the  axis 
can  easily  be  found. 

Let  CD  be  the  pendulum  in  question,  suspended  from  C,  the 
upper  extremity  of  the  vertical  axis  CD, 
and  let  the  ball  or  body  B,  by  revolving 
about  the  said  axis,  describe  the  circle  BE 
AH,  the  plane  of  which  is  parallel  to  the 
horizon  ;  it  is  proposed  to  assign  the  time 
of  description,  or  the  time  in  which  the  body 
B  performs  a  revolution  about  the  axis  CD, 
at  the'  distance  BD. 

Conceive  the  axis  CD  to  denote  the  weight 
of  the  revolving  body,  or  its  force  in  the  di- 
rection of  gravity;  then,  by  the  Compo- 
sition and  Resolution  of  Forces,  CB  will  denote  the  force  or 
tension  of  the  string  or  wire  that  retains  the  revolving  body  in 
the  direction  CB,  and  BD  the  force  tending  to  the  centre  of  the 
plane  of  revolution  at  D.  But,  by  the  general  laws  of  motion 
and  forces  previously  laid  down,  if  the  time  be  given,  the  space 
described  will  be  directly  proportional  to  the  force  ;  but,  by  the 
laws  of  gravity,  the  space  fallen  perpendicularly  from  rest,  in  one 
second  of  time,  is  g  =  16^  feet  ;  consequently  we  have  CD  :  BD  :  : 

1  P  1     T>"T\ 

16^2  :  —  12-  —  ,  the  space  described  towards  D  by  the  force  in  BD 
CD 

in  one  second.  Consequently,  by  the  laws  of  centripetal  forces,  the 
periodic  time,  or  the  time  of  the  body  revolving  in  the  circle  BEAH, 

V2-CD 
-  ,  where  *  =  3-1416,  the  circum- 


ference  of  a  circle  whose  diameter  is  unity  ;  or  putting  t  to  denote 
the  time,  and  expressing  the  height  CD  in  feet,  we  get  t  =  6-2832 

i       PT) 

*fe  —  ^-,  or,  by  reducing  the  expression  to  its  simplest  form,  it 

becomes  t  =  0-31986x/CD,  where  CD  must  be  estimated  in  inches, 
and  t  in  seconds.  Here  we  have  obtained  an  expression  of  great 
simplicity,  and  the  practical  rule  for  reducing  it  may  be  expressed 
in  words  as  follows  : 

RULE.  —  Multiply  the  square  root  of  the  height,  or  the  distance 
between  the  point  of  suspension  and  the  centre  of  the  plane  of  revo- 
lution, in  inches,  by  the  constant  fraction  0*31986,  and  the  product 
will  be  the  time  of  revolution  in  seconds. 

In  what  time  will  a  conical  pendulum  revolve  about  its  vertical 
axis,  supposing  the  distance  between  the  point  of  suspension  and 
the  centre  of  the  plane  of  revolution  to  be  39-1393  inches,  which  is 
the  length  of  a  simple  pendulum  that  vibrates  seconds  in  latitude 
51°  30'  ? 

The  square  root  of  39-1393  is  6-2561  ;  consequently,  by  the  rule, 


186 


THE    PRACTICAL    MODEL    CALCULATOR. 


we  have,  6-2561  x  0-31986  =  2-0011  seconds  for  the  time  of  revo- 
lution sought.  It  consequently  revolves  30  times  in  a  minute,  as  it 
ought  to  do  by  the  theory  of  the  simple  pendulum. 

By  reversing  the  process,  the  height  of  the  cone,  or  the  distance 
between  the  point  of  suspension  and  the  centre  of  the  plane  of  revo- 
lution, corresponding  to  any  given  time,  can  easily  be  ascertained ; 
for  we  have  only  to  divide  the  number  of  seconds  in  the  given  time 
by  the  constant  decimal  0-31986,  and  the  square  of  the  quotient 
will  be  the  required  height  in  inches.  Thus,  suppose  it  were  re- 
quired to  find  the  height  of  a  conical  pendulum  that  would  revolve 
30  times  in  a  minute.  Here  the  time  of  revolution  is  2  seconds  for 
60  -r-  30  =  2;  therefore,  by  division,  it  is  2  -r-  0-31986  =  6-2527, 
which,  being  squared,  gives  6-2527  =  39-0961  inches,  or  the  length 
of  a  simple  pendulum  that  vibrates  seconds  very  nearly.  In  all 
conical  pendulums  the  times  of  revolution,  or  the  periodic  times,  are 
proportional  to  the  square  roots  of  the  heights  of  the  cones.  This 
is  manifest,  for  in  the  foregoing  equation  of  the  periodic  time  the 
numbers  6-2832  and  386,j>r  12  X  32i,  are  constant  quantities,  con- 
sequently t  varies  as  >/CD. 

If  the  heights  of  the  cones,  or  the  distances  between  the  points 
of  suspension  and  the  centres  of  the  planes  of  revolution,  be  the 
same,  the  periodic  times,  or  the  times  of  revolution,  will  be  the 
same,  whatever  may  be  the  radii  of  the  circles  described  by  the  re- 


volving bodies.  This  will  be  clearly  understood  by  contemplating 
the  subjoined  diagram,  where  all  the  pendulums  Ca,"C6,  O,  Cd,  and 
Ce,  having  the  common  axis  CD,  will  revolve  in  the  same  time  ;  and 


THE    STEAM  ENGINE.  187 

if  they  are  all  in  the  same  vertical  plane  when  first  put  in  motion, 
they  will  continue  to  revolve  in  that  plane,  whatever  be  the  velocity, . 
so  long  as  the  common  axis  or  height  of  the  cone  remains  the  same. 
This  will  become  manifest,  if  we  conceive  an  inflexible  bar  or  rod 
of  iron  to  pass  through  the  centres  of  all  the  balls  as  well  as  the 
common  axis,  for  then  the  bar  and  the  several  balls  must  all  revolve 
in  the  same  time ;  but  if  any  one  of  them  should  be  allowed  to  rise 
higher,  its  velocity  would  be  increased ;  and  if  it  descends,  the  ve- 
locity will  be  decreased. 

Half  the  periodic  time  of  a  conical  pendulum  is  equal  to  the 
time  of  vibration  of  a  simple  pendulum,  the  length  of  which  is 
equal  to  the  axis  or  height  of  the  cone ;  that  is,  the  simple  pendu- 
lum makes  two  oscillations  or  vibrations  from  side  to  side,  or  it  * 
arrives  at  the  same  point  from  which  it  departed,  in  the  same  time 
that  the  conical  pendulum  revolves  about  its  axis.  The  space 
descended  by  a  falling  body  in  the  time  of  one  revolution  of  the 
conical  pendulum  is  equal  to  3-14162  multiplied  by  twice  the  height 
or  axis  of  the  cone.  The  periodic  time,  or  the  time  of  one  revo- 
lution is  equal  to  the  product  of  8-1416  v/  2  multiplied  by  the  time 
of  falling  through  the  height  of  the  cone.  The  weight  of  a  conical 
pendulum,  when  revolving  in  the  circumference  of  a  circle,  bears 
the  same  proportion  to  the  centrifugal  force,  or  its  tendency  to  fly 
off  in  a  straight  line,  as  the  axis  or  height  of  the  cone  bears  to  the 
radius  of  the  plane  of  revolution ;  consequently,  when  the  height 
of  the  cone  is  equal  to  the  radius  of  its  base,  the  centripetal  or 
centrifugal  force  is  equal  to  the  power  of  gravity. 

These  are  the  principles  on  which  the  action  of  the  conical  pen- 
dulum depends  ;  but  as  we  shall  hereafter  have  occasion  to  con- 
sider it  more  at  large,  we  need  not  say  more  respecting  it  in  this 
place.  Before  dismissing  the  subject,  however,  it  may  be  proper  to 
put  the  reader  in  possession  of  the  rules  for  calculating  the  posi- 
tion of  the  centre  of  oscillation  in  vibrating  bodies,  in  a  few  cases 
where  it  has  been  determined,  these  being  the  cases  that  are  of  the 
most  frequent  occurrence  in  practice. 

The  centre  of  oscillation  in  a  vibrating  body  is  that  point  in  the 
line  of  suspension,  in  which,  if  all  the  matter  of  the  system  were 
collected,  any  force  applied  there  would  generate  the  same  angular 
motion  in  a  given  time  as  the  same  force  applied  at  the  centre  of 
gravity.  The  centres  of  oscillation  for  several  figures  of  very  fre- 
quent use,  suspended  from  their  vertices  and  vibrating  flatwise,  are 
as  follow  : — 

In  a  right  line,  or  parallelogram,  or  a  cylinder  of  very  small 
diameter,  the  centre  of  oscillation  is  at  two-thirds  of  the  length 
from  the  point  of  suspension.  In  an  isosceles  triangle  the  centre 
of  oscillation  is  at  three-fourths  of  the  altitude.  In  a  circle  it 
is  five-fourths  of  the  radius.  In  the  common  parabola  it  is 
five-sevenths  of  its  altitude.  In  a  parabola  of  any  order  it  is 

3  n  +  ~\)  X  a^^U(^e>  wnere  n  denotes  the  order  of  the  figure. 


188  THE    PRACTICAL    MODEL    CALCULATOR. 

In  bodies  vibrating  laterally,  or  in  their  own  plane,  the  centres 
*of  oscillation  are  situated  as  follows ;  namely,  in  a  circle  the  centre 
of  oscillation  is  at  three-fourths  of  the  diameter ;  in  a  rectangle, 
suspended  at  one  of  its  angles,  it  is  at  two-thirds  of  the  diagonal  ; 
in  a  parabola,  suspended  by  the  vertex,  it  is  five-sevenths  of  the 
axis,  increased  by  one-third  of  the  parameter ;  in  a  parabola,  sus- 
pended by  the  middle  of  its  base,  it  is  four-sevenths  of  the  axis, 
increased  by  half  the  parameter;  in  the  sector  of  a  circle  it  is 
three  times  the  arc  of  the  sector  multiplied  by  the  radius,  and 
divided  by  four  times  the  chord  ;  in  a  right  cone  it  is  four-fifths  of 
the  axis  or  height,  increased  by  the  quotient  that  arises  when  the 
square  of  the  radius  of  the  base  is  divided  by  five  times  the  height ; 
in  a  globe  or  sphere  it  is  the  radius  of  the  sphere,  plus  the  length  of 
the  thread  by  which  it  is  suspended,  plus  the  quotient  that  arises 
when  twice  the  square  of  the  radius  is  divided  by  five  times  the  sum 
of  the  radius  and  the  length  of  the  suspending  thread.  In  all  these 
cases  the  distance  is  estimated  from  the  point  of  suspension,  and  since 
the  centres  of  oscillation  and  percussion  are  in  one  and  the  same 
point,  whatever  has  been  said  of  the  one  is  equally  true  of  the  other. 

THE   TEMPERATURE   AND   ELASTIC    FORCE    OF    STEAM. 

In  estimating  the  mechanical  action  of  steam,  the  intensity  of  its 
elastic  force  must  be  referred  to  some  known  standard  measure, 
such  as  the  pressure  which  it  exerts  against  a  square  inch  of  the 
surface  that  contains  it,  usually  reckoned  by  so  many  pounds 
avoirdupois  upon  the  square  inch.  The  intensity  of  the  elastic 
force  is  also  estimated  by  the  inches  in  height  of  a  vertical  column 
of  mercury,  whose  weight  is  equal  to  the  pressure  exerted  by  the 
steam  on  a  surface  equal  to  the  base  of  the  mercurial  column.  It 
may  also  be  estimated  by  the  height  of  a  vertical  column  of  water 
measured  in  feet ;  or  generally,  the  elastic  force  of  any  fluid  may 
be  compared  with  that  of  atmospheric  air  when  in  its  usual  state  of 
temperature  and  density ;  this  is  equal  to  a  column  of  mercury  30 
inches  or  2J  feet  in  height. 

When  the  temperature  of  steam  is  increased,  respect  being  had 
to  its  density,  the  elastic  force,  or  the  effort  to  separate  the  parts 
of  the  containing  vessel  and  occupy  a  larger  space,  is  also  increased  ; 
and  when  the  temperature  is  diminished,  a  corresponding  and  pro- 
portionate diminution  takes  place  in  the  intensity  of  the  emanci- 
pating effort  or  elastic  power.  It  consequently  follows  that  there 
must  be  some  law  or  principle  connecting  the  temperature  of  steam 
with  its  elastic  force ;  and  an  intimate  acquaintance  with  this  law, 
in  so  far  as  it  is  known,  must  be  of  the  greatest  importance  in  all 
our  researches  respecting  the  theory  and  the  mechanical  operations 
of  the  steam  engine. 

To  find  a  theorem,  by  means  of  which  it  may  be  ascertained  when 
a  general  law  exists,  and  to  determine  what  that  law  is,  in  cases 
where  it  is  known  to  obtain. — Suppose,  for  example,  that  it  is 
required  to  assign  the  nature  of  the  law  that  subsists  between  the 


THE    STEAM   ENGINE.  189 

temperature  of  steam  and  its  elastic  force,  on  tlie  supposition  that 
the  elasticity  is  proportional  to  some  power  of  the  temperature, 
and  unaffected  by  any  other  constant  or  co-efficient,  except  the 
exponent  by  which  the  law  is  indicated.  Let  E  and  e  be  any  two 
values  of  the  elasticity,  and  T,  £,  the  corresponding  temperatures 
deducted  from  observation.  It  is  proposed  to  ascertain  the  powers 
of  T  and  £,  to  which  E  and  e  are  respectively  proportional.  Let  n 
denote  the  index  or  exponent  of  the  required  power ;  then  by  the 
conditions  of  the  problem  admitting  that .  a  law  exists,  we  get, 

tn  € 

T" :  tn  : :  E  :  e ;  but  by  the  principles  of  proportion,  it  is  —  =  — ; 

and  if  this  be  expressed  logarithmically,  it  is  n  X  log.  —  =  log.  i 

1  E 

and  by  reducing  the  equation  in  respect  of  n,  it  finally  becomes 

_  log,  e  -  log.  E 
"  log.  t  -  log.  T' 

The  theorem  that  we  have  here  obtained  is  in  its  form  suffi- 
ciently simple  for  practical  application ;  it  is  of  frequent  occur- 
rence in  physical  science,  but  especially  so  in  inquiries  respecting 
the  motion  of  bodies  moving  in  air  and  other  resisting  media ;  and 
it  is  even  applicable  to  the  determination  of  the  planetary  motions 
themselves.  The  process  indicated  by  it  in  the  case  that  we  have 
chosen,  is  simply,  To  divide  the  difference  of  the  logarithms  of  the 
elasticities  by  the  difference  of  the  logarithms  of  the  corresponding 
temperatures,  and  the  quotient  will  express  that  power  of  the  tempe- 
rature to  which  the  elasticity  is  proportional. 

Take  as  an  example  the  following  data  : — In  two  experiments  it 
was  found  that  when  the  temperature  of  steam  was  250*3  and 
343*6  degrees  of  Fahrenheit's  scale,  the  corresponding  elastic 
forces  were  59*6  and  238*4  inches  of  the  mercurial  column  respec- 
tively. From  these  data  it  is  required  to  determine  the  law  which 
connects  the  temperature  with  the  elastic  force  on  the  supposition 
that  a  law  does  actually  exist  under  the  specified  conditions.  The 
process  by  the  rule  is  as  follows : 

Greater  temperature,  343*6  log.  2*5352941 

Lesser  temperature,  250*3 log.  2*3984608 

Remainder =  0*1368333 

Greater  elastic  force,  238*4  log.  2*3773063 

Lesser  elastic  force,  59-6  log.  1*7752463 

Remainder  =  0*6020600 

Let  the  second  of  these  remainders  be  divided  by  the  first,  as 
directed  in  the  rule,  and  we  get  n  =  6020600  -f- 1368333  =  4*3998, 
the  exponent  sought.  Consequently,  by  taking  the  nearest  unit, 
for  the  sake  of  simplicity,  we  shall  have,  according  to  this  result, 
the  following  analogy,  viz. : 

T4'4 :  £4'4 : :  E  :  e  ; 


190  THE    PRACTICAL   MODEL   CALCULATOR. 

that  is,  the  elasticities  are  proportional  to  the  4'4  power  of  the 
temperatures  very  nearly. 

Now  this  law  is  rigorously  correct,  as  applied  to  the  particular 
cases  that  furnished  it ;  for  if  the  two  temperatures  and  one  elas- 
ticity be  given,  the  other  elasticity  will  be  found  as  indicated  by 
the  above  analogy ;  or  if  the  two  elasticities  and  one  temperature 
be  given,  the  other  temperature  will  be  found  by  a  similar  process. 
It  by  no  means  follows,  however,  that  the  principle  is  general,  nor 
could  we  venture  to  affirm  that  the  exponent  here  obtained  will 
accurately  represent  the  result  of  any  other  experiments  than 
those  from  which  it  is  deduced,  whether  the  temperature  be  higher 
or  lower  than  that  of  boiling  water ;  but  this  we  learn  from  it,  that 
the  index  which  represents  the  law  of  elasticity  is  of  a  very  high 
order,  and  that  the  general  equation,  whatever  its  form  may  be, 
must  involve  other  conditions  than  those  which  we  have  assumed  in 
the  foregoing  investigation.  The  theorem,  however,  is  valuable  to 
practical  men,  not  only  as  being  applicable  to  numerous  other 
branches  of  mechanical  inquiry,  but  as  leading  directly  to  the 
methods  by  which  some  of  the  best  rules  have  been  obtained  for 
calculating  the  elasticity  of  steam,  when  in  contact  with  the  liquid 
from  which  it  is  generated. 

We  now  proceed  to  apply  our  formula  to  the  determination  of  a 
general  law,  or  such  as  will  nearly  represent  the  class  of  experi- 
ments on  which  it  rests ;  and  for  this  purpose  we  must  first  assign 
the  limits,  and  then  inquire  under  what  conditions  the  limitations 
take  place,  for  by  these  limitations  we  must  in  a  great  measure  be 
guided  in  determining  the  ultimate  form  of  the  equation  which 
represents  the  law  of  elasticity. 

The  limits  of  elasticity  will  be  readily  assigned  from  the  follow- 
ing considerations,  viz.  :  In  the  first  place,  it  is  obvious  that  steam 
cannot  exist  when  the  cohesive  attraction  of  the  particles  is  of 
greater  intensity  than  the  repulsive  energy  of  the  caloric  or  matter 
of  heat  interposed  between  them ;  for  in  this  case,  the  change  from 
an  elastic  fluid  to  a  solid  may  take  place  without  passing  through 
the  intermediate  stage  of  liquidity :  hence  we  infer  that  there  must 
be  a  temperature  at  which  the  elastic  force  is  nothing,  and  this 
temperature,  whatever  may  be  its  value,  corresponds  to  the  lower 
limit  of  elasticity.  The  higher  limit  will  be  discovered  by  similar 
considerations,  for  it  must  take  place  when  the  density  of  steam  is 
the  same  as  that  of  water,  which  therefore  depends  on  the  modulus 
of  elasticity  of  water.  The  modulus  of  elasticity  of  any  substance 
is  the  measure  of  its  elastic  force ;  that  of  water  at  60°  of  tempe- 
rature is  22,100  atmospheres.  Thus,  for  instance,  suppose  a  given 
quantity  of  water  to  be  confined  in  a  close  vessel  which  it  exactly 
fills,  and  let  it  be  exposed  to  a  high  degree  of  temperature,  then 
it  is  obvious  that  in  this  state  no^steam  would  be  produced,  and  the 
force  which  is  exerted  to  separate  the  parts  of  the  vessel  is  simply 
the  expansive  force  of  compressed  water ;  we  therefore  have  the 
following  proportion.  As  the  expanded  volume  of  water  is  to  the 


THE    STEAM    ENGINE.  191 

quantity  of  expansion,  so  is  the  modulus  of  elasticity  of  water  to 
the  elastic  force  of  steam  of  the  same  density  as  water. 

Having  therefore  assigned  the  limits  beyond  which  the  elastic 
force  of  steam  cannot  reach.,  we  shall  now  proceed  to  apply  the 
principle  of  our  formula  to  the  determination  of  the  general  law 
which  connects  .the  temperature  with  the  elastic  force  ;  and  for  this 
purpose,  in  addition  to  the  notation  which  we  have  already  laid 
down,  let  c  denote  some  constant  quantity  that  affects  the  elasticity, 
and  d  the  temperature  at  which  the  elasticity  vanishes  ;  then  since 
this  temperature  must  be  applied  subtractively,  we  have  from  the 
foregoing  principle,  c  E  =  (T  —  g)n,  and  c  e  —  (t  —  8)n.  From 
either  of  these  equations,  therefore,  the  constant  quantity  c  can 
be  determined  in  terms  of  the  rest  when  they  are  known ;  thus  we 

have  c  =  ^ — — — '—,  and  c  —  ± L,   and  by  comparing  these 

E  e 

two  independent  values  of  e,  the  value  of  n  becomes  known ;  for 

(T  —  8)"        (t  —  8)» 

* — _ — '—  —  \ L,    and  consequently 

E  e 

log.  e  —  log.  E 


log.  (t  -  8)  -  log.  (T  -  8). 
In  this  equation  the  value  of  the  symbol  8  is  unknown  ;  in  order 
therefore  to  determine  it,  we  must  have  another  independent 
expression  for  the  value  of  n ;  and  in  order  to  this,  let  the  elasti- 
cities E  and  e  become  E'  and  ef  respectively ;  while  the  corre- 
sponding temperatures  T  and  t  assume  the  values  T'  and  t' ;  then 

/rjV    .An  /./    «\ 

by  a  similar  process  to  the  above,  we  get  ' —       '    =  V — J,  and 

Jb/  e' 

_  log.  e'  -  log.  E' (B) 

"  log.  («'  -  8)  - :iog.  (T'  -  8)  ; 

Let  the  equations  (A)  and  (B)  be  compared  with  each  other,  and 
we  shall  then  have  an  expression  involving  only  the  unknown 
quantity  5,  for  it  must  be  understood  that  the  several  temperatures 
with  their  corresponding  elasticities  are  to  be  deduced  from  experi- 
ment ;  and  in  consequence,  the  law  that  we  derive  from  them  must 
be  strictly  empirical ;  thus  we  have 

log.  e  —  log.  E.          _  log.  e'  —  log.  E  /p\ 

log.  (t  —  8)  —  log.  (T  —  8)  ~~  logT(£'  —  6)~^log.7T'  —  6) 

We  have  no  direct  method  of  reducing  expressions  of  this  sort, 
and  the  usual  process  is  therefore  by  approximation,  or  by  the  rule 
of  trial  and  error,  and  it  is  in  this  way  that  the  value  of  the  quan- 
tity 8  must  be  found ;  and  for  the  purpose  of  performing  the  reduc- 
tion, we  shall  select  experiments  performed  with  great  care,  and 
may  consequently  be  considered  as  representing  the  law  of  elas- 
ticity with  very  great  nicety. 

T  =  212-0  Fahrenheit  E  =    29-8  inches  of  mercury. 
t  =  250-3  e  =    59-6 

T;=  293-4  E'=  119-2 

£'=343-6  e'=  238-4 


192  THE   PRACTICAL   MODEL   CALCULATOR. 

Therefore,  by  substituting  these  numbers  in  "equation  (C),  and 
making  a  few  trials,  we  find  that  8  =  —  50°,  and  substituting  this 
in  either  of  the  equations  (A)  or  (B),  we  get  n  =  5-08  ;  and 
finally,  by  substituting  these  values  of  &  and  n  in  either  of  the 
expressions  for  the  constant  quantity  <?,  we  get  c  =  64674780000, 
the  5*08  root  of  which  is  134-27  very  nearly ;  hence  we  have 


134-27 

"Where  the  symbol  F  denotes  generally  the  elastic  force  of  the 
steam  in  inches  of  mercury,  and  t  the  corresponding  temperature 
in  degrees  of  Fahrenheit's  thermometer,  the  logarithm  of  the 
denominator  of  the  fraction  is  2-1279717,  which  may  be  used-  as  a 
constant  in  calculating  the  elastic  force  corresponding  to  any  given 
temperature.  We  have  thus  discovered  a  rule  of  a  very  simple 
form  ;  it  errs  in  defect ;  but  this  might  have  been  remedied  by 
assuming  two  points  near  one  extremity  of  the  range  of  experi- 
ment, and  two  points  near  the  other  extremity  ;  and  by  substi- 
tuting the  observed  numbers  in  equation  (C),  different  constants 
and  a  more  correct  exponent  would  accordingly  have  been  obtained. 
Mr.  Southern  has,  by  pursuing  a  method  somewhat  analogous  to 
that  which  is  here  described,  found  his  experiments  to  be  very 
nearly  represented  by 


F=-  /*  +  51-3  \ 
I  135-767  J 


But  even  here  the  formula  errs  in  defect,  for  he  has  found  it 
necessary  to  correct  it  by  adding  the  arbitrary  decimal  0-1;  and 
thus  modified,  it  becomes 


Our  own  formula  may  also  be  corrected  by  the  application  of 
some  arbitrary  constant  of  greater  magnitude  ;  but  as  our  motive 
for  tracing  the  steps  of  investigation  in  the  foregoing  case  was  to 
exemplify  the  method  of  determining  the  law  of  elasticity,  our  end 
is  answered  ;  for  we  consider  it  a  very  unsatisfactory  thing  merely 
to  be  put  in  possession  of  a  formula  purporting  to  be  applicable  to 
some  particular  purpose,  without  at  the  same  time  being  put  in 
possession  of  the  method  by  which  that  formula  was  obtained,  and 
the  principles  on  which  it  rests.  Having  thus  exhibited  the  prin- 
ciples and  the  method  of  reduction,  the  reader  will  have  greater 
confidence  as  regards  the  consistency  of  the  processes  that  he  may 
be  called  upon  to  perform.  The  operation,  implied  by  equation  (E) 
may  be  expressed  in  words  as  follows  :  — 

RULE.  —  To  the  given  temperature  in  degrees  of  Fahrenheit's 
thermometer  add  51-3  degrees  and  divide  the  sum  by  135-767  ;  to 
the  5-13  power  of  the  quotient  add  the  constant  fraction  ^,  and 
the  sum  will  be  the  elastic  force  in  inches  of  mercury. 


THE   STEAM    ENGINE.  193 

The  process  here  described  is  that  which  is  performed  by  the 
rules  of  common  arithmetic ;  but  since  the  index  is  affected  by  a 
fraction,  it  is  difficult  to  perform  in  that  way :  we  must  therefore 
have  recourse  to  logarithms  as  the  only  means  of  avoiding  the  diffi- 
culty. The  rule  adapted  to  these  numbers  is  as  follows : — 

RULE  FOR  LOGARITHMS. — To  the  given  temperature  in  degrees 
of  Fahrenheit's  thermometer  add  51 -3  degrees ;  then,  from  the 
logarithm  of  the  sum  subtract  2-1327940  or  the  logarithm  of 
135-767,  the  denominator  of  the  fraction ;  multiply  the  remainder 
by  the  index  5-13,  and  to  the  natural  number  answering  to  the 
sum  add  the  constant  fraction  ^ ;  the  sum  will  be  the  elastic  force 
in  inches  of  mercury. 

If  the  temperature  of  steam  be  250-3  degrees  as  indicated  by 
Fahrenheit's  thermometer,  what  is  the  corresponding  elastic  force 
in  inches  of  mercury  ? 
By  the  rule  it  is  250-3  +  51-3  =  301-6      log.  2-4794313 

constant  den.  =  135-767  log.  2-1327940  subtract 

remainder  =  0-3466373 

31-5  inverted 

17331865 
346637 
103991 

natural  number  60-013  log.  1-7782493 

If  this  be  increased  by  ^,  we  get  60-113  inches  of  mercury  for 
the  elastic  force  of  steam  at  250-3  degrees  of  Fahrenheit. 

By  simply  reversing  the  process  or  transposing  equation  (E),  the 
temperature  corresponding  to  any  given  elastic  force  can  easily  be- 
found ;  the  transformed  expression  is  as  follows,  viz. : 

t  =  135-767  (F  -  0-1)^  -  51-3    ....  (F). 

Since,  in  consequence  of  the  complicated  index,  the  process  of 
calculation  cannot  easily  be  performed  by  common  arithmetic,  it  is- 
needless  to  give  a  rule  for  reducing  the  equation  in  that  way ;  we 
shall  therefore  at  once  give  the  rule  for  performing  the  process  by 
logarithms. 

RULE. — From  the  given  elastic  force  in  inches  of  mercury,  sub- 
tract the  constant  fraction  0*1 ;  divide  the  logarithm  of  the  remain- 
der by  5-13,  and  to  the  quotient  add  the  logarithm  2-1327940  ;  find 
the  natural  number  answering  to  the  sum  of  the  logarithms,  and 
from  the  number  thus  found  subtract  the  constant  51*3,  and  the 
remainder  will  be  the  temperature  sought. 

Supposing  the  elastic  force  of  steam  or  the  vapour  of  water  to 
be  equivalent  to  the  weight  of  a  vertical  column  of  mercury,  the 
height  of  which  is  238-4  inches ;  what  is  the  corresponding  tem- 
perature in  degrees  of  Fahrenheit's  thermometer  ? 

Here,  by  proceeding  as  directed  in  the  rule,  we  have  238-4  —0*1  = 
B  13 


194  THE    PRACTICAL   MODEL    CALCULATOR. 

238-3,  and  dividing  the  logarithm  of  this  remainder  by  the  con- 

stant exponent  5'13,  we  get 

log.  238-3  H-  5-13       =      2-3771240  +  5-13  =  0-4633770 
constant  co-efficient     =135-767      -      -      log.  2-1327940  add 

natural  number  =394-61      -     -     -     log.  2-5961710  sum 

constant  temperature  =    51*3  subtract 

required  temperature  =  343-31  degrees  of  Fahrenheit's  ther- 
mometer. 

The  temperature  by  observation  is  343-6  degrees,  giving  a  differ- 
ence of  only  0-29  of  a  degree  in  defect.  For  low  temperature  or 
low  pressure  steam,  that  is,  steam  not  exceeding  the  simple  pres- 
sure of  the  atmosphere,  M.  Pambour  gives 


p  =  0-04948  +    ig 


In  which  equation  the  symbol  p  denotes  the  pressure  in  pounds 
avoirdupois  per  square  inch,  and  t  the  temperature  in  degrees  of 
Fahrenheit's  thermometer.  When  this  expression  is  reduced  in 
reference  to  temperature,  it  is 

£  =  155-7256  Q?-  0-04948)  ™-51-3    ....  (H). 

The  formula  of  Tredgold  is  well  known.  The  equation,  in  its 
original  form,  is 

177/*  =  t  +  100  .     .     .     .  (I)  : 

where  /denotes  the  elastic  force  of  steam  in  inches  of  mercury, 
and  t  the  temperature  in  degrees  of  Fahrenheit's  thermometer. 
The  same  formula,  as  modified  and  corrected  by  M.  Millet,  becomes 

179-0773/*  =  t  -f  103  .     .     .     .  (K). 

Dr.  Young  of  Dublin  constructed  a  formula  which  was  adapted 
to  the  experiments  of  his  countryman  Dr.  Dalton  :  it  assumed  a 
form  sufficiently  simple  and  elegant  ;  it  is  thus  expressed  — 
/  =  (i  +  0.0029  1)7  .     .     .     .  (L)  : 

where  the  symbol  f  denotes  the  elastic  force  of  steam  expressed  in 
atmospheres  of  30  inches  of  mercury,  and  t  the  temperature  in 
degrees  estimated  above  212  of  Fahrenheit.  This  formula  is  not 
applicable  in  practice,  especially  in  high  temperatures,  as  it  deviates 
very  widely  and  rapidly  from  the  results  of  observation:  it  'is 
chiefly  remarkable  as  being  made  the  basis  of  a  numerous  class  of 
theorems  somewhat  varied,  but  of  a  more  correct  and  satisfactory 
character.  The  Commission  of  the  French  Academy  represented 
their  experiments  by  means  of  a  formula  constructed  on  the  same 
principles  :  it  is  thus  expressed  — 

/  =  (1  -f  0-7153  1)*  .     .     .     .  (M)  : 

where  /denotes  the  elastic  force  of  the  steam  expressed  in  atmo- 
spheres of  0-76  metres  or  29-922  inches  of  mercury,  and  t  the  tern- 


THE   STEAM   ENGINE.  195 

perature  estimated  above  100  degrees  of  the  centigrade  thermo- 
meter ;  but  when  the  same  formula  is  so  transformed  as  to  be 
expressed  in  the  usual  terms  adopted  in  practice,  it  is 

p  =  (0-2679  +  0-0067585  t)s  .     .     .     .   (N): 

where  p  is  the  pressure  in  pounds  per  square  inch,  and  t  the  tem- 
perature in  degrees  of  Fahrenheit's  scale,  estimated  above  212  or 
simple  atmospheric  pressure. 

The  committee  of  the  Franklin  Institute  adopted  the  exponent 
6,  and  found  it  necessary  to  change  the  constant  0-0029  into 
0*00333 ;  thus  modified,  they  represented  their  experiments  by 
the  equation 

p  =  (tf-460467  +  0-00521478  t)6  .     .     .     .  (0). 

By  combining  Dr.  Dalton's  experiments  with  the  mean  between 
those  of  the  French  Academy  and  the  Franklin  Institute,  we  obtain 
the  following  equations,  the  one  being  applicable  for  temperatures 
below  212  degrees,  and  the  other  for  temperatures  above  that 
point  as  far  as  50  atmospheres.  Thus,  for  low  pressure  steam, 
that  is,  for  steam  of  less  temperature  than  212,  it  is 


and  for  steam  above  the  temperature  of  212,  it  is 

/-(Mnrarr-  •  •  •  (<«• 

In  consequence  therefore  of  the  high  and  imposing  authority 
from  which  these  formulas  are  deduced,  we  shall  adopt  them  in  all 
our  subsequent  calculations  relative  to  the  steam  engine ;  and  in 
order  to  render  their  application  easy  and  familiar,  we  shall  trans- 
late them  into  rules  in  words  at  length,  and  illustrate  them  by  the 
resolution  of  appropriate  numerical  examples;  and  for  the  sake  of 
a  systematic  arrangement,  we  think  proper  to  branch  the  subject 
into  a  series  of  problems,  as  follows : 

The  temperature  of  steam  being  given  in  degrees  of  Fahrenheit's 
thermometer,  to  find  the  corresponding  elastic  force  in  inches  of 
mercury. — The  problem,  as  here  propounded,  is  resolved  by  one  or 
other  of  the  last  two  equations,  and  the  process  indicated  by  the 
arrangement  is  thus  expressed : — 

RULE. — To  the  given  temperature  expressed  in  degrees  of 
Fahrenheit's  thermometer,  add  the  constant  temperature  175 ;  find 
the  logarithm  answering  to  the  sum,  from  which  subtract  the  con- 
stant 2-587711 ;  multiply  the  remainder  by  the  index  7-71307,  and 
the  product  will  be  the  logarithm  of  the  elastic  force  in  atmospheres 
of  3$  inches  of  mercury  when  the  given  temperature  is  less  than 
212  degrees.  But  when  the  temperature  is  greater  than  212, 
increase  it  by  121 ;  then,  from  the  logarithm  of  the  temperature 
thus  increased,  subtract  the  constant  logarithm  2*522444,  multiply 
the  remainder  by  the  exponent  6 '42,  and  the  product  will  be  thf* 


196  THE   PRACTICAL   MODEL   CALCULATOR. 

logarithm  of  the  elastic  force  in  atmospheres  of  30  inches  of  mer- 
cury ;  which  being  multiplied  by  30  will  give  the  force  in  inches, 
or  if  multiplied  by  14-76  the  result  will  be  expressed  in  pounds 
avoirdupois  per  square  inch. 

When  steam  is  generated  under  a  temperature  of  187  degrees  of 
Fahrenheit's  thermometer,  what  is  its  corresponding  elastic  force  in 
atmospheres  of  30  inches  of  mercury  ? 

In  this  example,  the  given  temperature  is  less  than  212  degrees : 
it  will  therefore  be  resolved  by  the  first  clause  of  the  preceding 
rule,  in  which  the  additive  constant  is  175 ;  hence  we  get 

187  +  175  =  362. ..log.  2-558709 
Constant  divisor  =  387. ..log.  2-587711  subtract 

9-970998  x  7-71307  =  9-773393 

And  the  corresponding  natural  number  is  0*5934  atmospheres,  or 
17*802  inches  of  mercury,  the  elastic  force  required,  or  if  expressed 
in  pounds  per  square  inch,  it  is  0-5934  X  14-76  =  8*76  Ibs.  very 
nearly.  If  the  temperature  be  250  degrees  of  Fahrenheit,  the  pro- 
cess is  as  follows : 

250  +  121  =  871.-.log.  2-569374 
Constant  divisor  =  333... log.  2-522444  subtract 

0-046930  x  6*42  =  0*301291 

And  the  corresponding  natural  number  is  2*0012  atmospheres,  or 
60*036  inches  of  mercury,  and  in  pounds  per  square  inch  it  is 
2-0012  x  14-76  =  29-54  Ibs.  very  nearly. 

It  is  sometimes  convenient  to  express  the  results  in  inches  of 
mercury,  without  a  previous  determination  in  atmospheres,  and  for 
this  purpose  the  rule  is  simply  as  follows : 

RULE. — Multiply  the  given  temperature  in  degrees  of  Fahren- 
heit's thermometer  by  the  constant  coefficient  1-5542,  and  to  the 
product  add  the  constant  number  271*985;  then  from  the  loga- 
rithm of  the  sum  subtract  the  constant  logarithm  2-587711,  and 
multiply  the  remainder  by  the  exponent  7*71307  ;  the  natural  num- 
ber answering  to  the  product,  considered  as  a  logarithm,  will  give 
the  elastic  force  in  inches  of  mercury.  This  answers  to  the  case 
when  the  temperature  is  less  than  212  degrees;  but  when  it  is 
above  that  point  proceed  as  follows  : 

Multiply  the  given  temperature  in  degrees  of  Fahrenheit's  ther- 
mometer by  the  constant  coefficient  1*69856,  and  to  the  product  add 
the  constant  number  205*526 ;  then  from  the  logarithm  of  the  sum 
subtract  the  constant  logarithm  2*522444,  and  multiply  the  re- 
mainder by  the  exponent  6*42 ;  the  natural  number  answering  to 
the^  product  considered  as  a  logarithm,  will  give  the  elastic  force 
in  inches  of  mercury.  Take,  for  example,  the  temperatures  as 
assumed  above,  and  the  process,  according  to  the  rule,  is  as  fol- 
lows: 


THE   STEAM   ENGINE.  197 

» 

187  x  1-5542  =  290-6354 
Constant  =  271'985  add 

Sum  =  562-6204...1og.  2-750216 
Constant  =  387 log.  2-587711  subtract 

0162505  x  7-71307  =  1-253408 

And  the  natural  number  answering  to  this  logarithm  is  17*923  inches 
of  mercury.  By  the  preceding  calculation  the  result  is  17*802 ; 
the  slight  difference  arises  from  the  introduction  of  the  decimal  con- 
stants, which  in  consequence  of  not  terminating  at  the  proper  place 
are  taken  to  the  nearest  unit  in  the  last  figure,  but  the  process  is 
equally  true  notwithstanding.  For  the  higher  temperature,  we  get 
250  x  1-69856  =  424-640 

Constant  =  205-526  add 

Sum  =  630.166 log.  2-799456 

Constant  =  333 log.  2-522444  subtract 

0-277011  x  6-42  =  1-778410 

And  the  natural  number  answering  to  this  logarithm  is  60-036 
inches  of  mercury,  agreeing  exactly  with  the  result  obtained  as 
above. 

It  is  moreover  sometimes  convenient  to  express  the  force  of  the 
steam  in  pounds  per  square  inch,  without  a  previous  determination 
in  atmospheres  or  inches  of  mercury;  and  when  the  equations  are 
modified  for  that  purpose,  they  supply  us  with  the  following  process, 
viz.: 

Multiply  the  given  temperature  by  the  constant  coefficient 
1-41666,  and  to  the  product  add  the  constant  number  247*9155; 
then,  from  the  logarithm  of  the  sum  subtract  the  constant  logarithm 
2-587711,  and  multiply  the  remainder  by  the  index  7*71307  ;  the 
natural  number  answering  to  the  product  will  give  the  pressure  in 
pounds  per  square  inch,  when  the  temperature  is  less  than  212  de- 
grees ;  but  for  all  greater  temperatures  the  process  is  as  follows : 

Multiply  the  given  temperature  by  the  constant  coefficient 
1-5209,  and  to  the  product  add  the  constant  number  184-0289 ; 
then,  from  the  logarithm  of  the  sum  subtract  the  constant  logarithm 
2-522444,  and  multiply  the  remainder  by  the  exponent  6-42 ;  the 
natural  or  common  number  answering  to  the  product,  will  express 
the  force  of  the  steam  in  pounds  per  square  inch.  If  any  of  these 
results  be  multiplied  by  the  decimal  0-7854,  the  product  will  be  the 
corresponding  pressure  in  pounds  per  circular  inch.  Taking,  there- 
fore, the  temperatures  previously  employed,  the  operation  is  as 
follows : 
187  x  1-41666  =  264-9155 

Constant  =  247-9155  add 

Sum  =  512.8310.log.  2-709974 
Constant  =  387 log.  2*587711  subtract 

0-122263  x  7-71307  =  0*942656 

E2 


198  THE    PRACTICAL   MODEL   CALCULATOR. 

And  the  number  answering  to  this  logarithm  is  8*763  Ibs.  per  square 
inch,  and  8-763  X  0-7854  =  6-8824  Ibs.  per  circular  inch,  the  pro- 
portion in  the  two  cases  being  as  1  to  0-7554.  Again,  for  the 
higher  temperature,  it  is 

250  x  1-5209  =  380-2250 

Constant  =  184-0289  add 

Sum  =  564-2539 log.  2-751475 

Constant  =  333 log.  2-522444  subtract 

0-229031  x  6-42  =  1-470279 

And  the  number  answering  to  this  logarithm  is  29-568  Ibs.  per 
square  inch,  or  29568  X  0-7854  =  23-2226  Ibs.  per  circular  inch. 

We  have  now  to  reverse  the  process,  and  determine  the  tempera- 
taire  corresponding  to  any  given  power  of  the  steam,  and  for  this 
purpose  we  must  so  transpose  the  formulas  (P)  and  (Q),  as  to  express 
the  temperature  in  terms  of  the  elastic  force,  combined  with  given 
constant  numbers ;  but  as  it  is  probable  that  many  of  our  readers 
would  prefer  to  see  the  theorems  from  which  the  rules  are  deduced, 
we  here  subjoin  them. 

For  the  lower  temperature,  or  that  which  does  not  exceed  the 
temperature  of  boiling  water,  we  get 

t  =  249/^-175  ....  (R). 

Where  t  denotes  the  temperature  in  degrees  of  Fahrenheit's  ther- 
mometer, and /the  elastic  force  in  inches  of  mercury,  less  than  30 
inches,  or  one  atmosphere ;  but  when  the  elastic  force  is  greater 
than  one  atmosphere,  the  formula  for  the  corresponding  temperature 

is  as  follows : 

i 

t  =  196/6'42  -  121  ....  (S). 

In  the  construction  of  these  formulas,  we  have,  for  the  sake  of 
simplicity,  omitted  the  fractions  that  obtain  in  the  coefficient  off; 
for  since  they  are  very  small,  the  omission  will  not  produce  an  error 
of  any  consequence ;  indeed,  no  error  will  arise  on  this  account,  as 
we  retain  the  correct  logarithms,  a  circumstance  that  enables  the 
computer  to  ascertain  the  true  value  of  the  coefficients  whenever  it 
is  necessary  so  to  do  ;  but  in  all  cases  of  actual  practice,  the  results 
derived  from  the  integral  coefficients  will  be  quite  sufficient.  The 
rule  supplied  by  the  equations  (R)  and  (S)  is  thus  expressed : 

When  the  elastic  force  is  less  than  the  pressure  of  the  atmosphere, 
that  is,  less  than  30  inches  of  the  mercurial  column, — 

RULE. — Divide  the  logarithm  of  the  given  elastic  force  in  inches 
of  mercury,  by  the  constant  index  7'71307,  and  to  the  quotient  add 
the  constant  logarithm  2-396204;  then  from  the  common  oj:  natural 
number  answering  to  the  sum,  subtract  the  constant  temperature 
175  degrees,  and  the  remainder  will  be  the  temperature  sought  in 
degrees  of  Fahrenheit's  thermometer.  But  when  the  elastic  force 
exceeds  30  inches,  or  one  atmosphere,  the  following  rule  applies : 


THE    STEAM    ENGINE.  199 

Divide  the  logarithm  of  the  given  elastic  force  in  inches  of  mer- 
cury by  the  constant  index  6-42,  and  to  the  quotient  add  the  con- 
stant logarithm  2-292363 :  then,  from  the  natural  number  answer- 
ing to  the  sum  subtract  the  constant  temperature  121  degrees,  and 
the  remainder  will  be  the  temperature  sought.  Similar  rules  might 
be  constructed  for  determining  the  temperature,  when  the  pressure 
in  pounds  per  square  inch  is  given ;  but  since  this  is  a  less  useful 
case  of  the  problem,  we  have  thought  proper  to  omit  it.  We  there- 
fore proceed  to  exemplify  the  above  rules,  and  for  this  purpose  we 
shall  suppose  the  pressure  in  the  two  cases  to  be  equivalent  to  the 
weight  of  19  and  60  inches  of  mercury  respectively.  The  operations 
will  therefore  be  as  follows  : 

Log.  19  --  7-71307  =  1-278754  -r-  7*71307  =  0-165791 
Constant  coefficient  =  249 log.  2-396204  add 

Natural  number  =  364-75 log.  2-561994 

Constant  temperature  =  175      subtract 

Required  temperature  =  189*75  degrees  of  Fahrenheit's  scale. 
For  the  higher  elastic  force  the  operation  is  as  follows : 

Log.  60  -*-  6-42  =  1-778151  -r-  6-42  =  0-276969 
Constant  coefficient  =  196 log.  2-292363  add 

Natural  number  =  370-97 log.  2-569332 

Constant  temperature  =  121      subtract 

Required  temperature  =  249-97  degrees  of  Fahrenheit's  scale. 

All  the  preceding  results,  as  computed  by  our  rules,  agree  as 
nearly  with  observation  as  can  be  desired :  but  they  have  all  been 
obtained  on  the  supposition  that  the  steam  is  in  contact  with  the 
liquid  from  which  it  is  generated ;  and  in  this  case  it  is  evident 
that  the  steam  must  always  attain  an  elastic  force  corresponding  to 
the  temperature ;  and  in  accordance  to  any  increase  of  pressure, 
supposing  the  temperature  to  remain  the  same,  a  quantity  of  it 
corresponding  to  the  degree  of  compression  must  simply  be  condensed 
into  water,  and  in  consequence  will  leave  the  diminished  space 
occupied  by  steam  of  the  original  degree  of  tension ;  or  otherwise 
to  express  it,  if  the  temperature  and  pressure  invariably  correspond 
with  each  other,  it  is  impossible  to  increase  the  density  and  elas- 
ticity of  the  steam  except  by  increasing  the  temperature  at  the  same 
time  ;  and,  contrariwise,  the  temperature  cannot  be  increased  with- 
out at  the  same  time  increasing  the  elasticity  and  density.  This 
being  admitted,  it  is  obvious  that  under  these  circumstances  the 
steam  must  always  maintain  its  maximum  of  pressure  and  density : 
but  if  it  be  separated  from  the  liquid  that  produces  it,  and  if  its 
temperature  in  this  case  be  increased,  it  will  be  found  not  to  possess 
a  higher  degree  of  elasticity  than  a  volume  of  atmospheric  air  simi- 
larly confined,  and  heated  to  the  same  temperature.  Under  this 
'new  condition,  the  state  of  maximum  density  and  elasticity  ceases ; 
for  it  id  qbvious  that  since  no  water  is  present,  there  cannot  be  any 


200  THE   PRACTICAL   MODEL   CALCULATOR. 

more  steam  generated  by  an  increase  of  temperature ;  and  conse- 
quently the  force  of  the  steam  is  only  that  which  confines  it  to  its 
original  bulk,  and  is  measured  by  the  effort  which  it  exerts  to  ex- 
pand itself.  Our  next  object,  therefore,  is  to  inquire  what  is  the 
law  of  elasticity  of  steam  under  the  conditions  that  we  have  here 
specified. 

The  specific  gravity  of  steam,  its  density,  and  the  volume  which 
it  occupies  at  different  temperatures,  have  been  determined  by  ex- 
periment with  very  great  precision ;  and  it  has  also  been  ascertained 
that  the  expansion  of  vapour  by  means  of  heat  is  regulated  by  the 
same  laws  as  the  expansion  of  the  other  gases,  viz.  that  all  gases 
expand  from  unity  to  1-375  in  bulk  by  180  degrees  of  temperature; 
and  again,  that  steam  obeys  the  law  discovered  by  Boyle  and  Mari- 
otte,  contracting  in  volume  in  proportion  to  the  degree  of  pressure 
which  it  sustains.  We  have  therefore  to  inquire  what  space  a  given 
quantity  of  water  converted  into  steam  will  occupy  at  a  given  pres- 
sure ;  and  from  thence  we  can  ascertain  the  specific  gravity,  density, 
and  volume  at  all  other  pressures. 

When  a  gas  or  vapour  is  submitted  to  a  constant  pressure,  the 
quantity  which  it  expands  by  a  given  rise  of  temperature  is  calcu- 
lated by  the  following  theorem, 

^+459\         (T) 


t  +  4597 

where  t  and  tf  are  the  temperatures,  and  v,  vf  the  corresponding 
volumes  before  and  after  expansion;  hence  this  rule. 

RULE. — To  each  of  the  temperatures  before  and  after  expansion, 
add  the  constant  experimental  number  459  ;  divide  the  greater  sum 
by  the  lesser,  and  multiply  the  quotient  by  the  volume  at  the  lower 
temperature,  and  the  product  will  give  the  expanded  volume. 

If  "the  volume  of  steam  at  the  temperature  of  212  degrees  of  Fah- 
renheit be  1711  times  the  bulk  of  the  water  that  produces  it,  what 
will  be  its  volume  at  the  temperature  of  250-3  degrees,  supposing 
the  pressure  to  be  the  same  in  both  cases  ? 

Here,  by  the  rule,  we  have  212  -f  459  =  671,  and  250-3  +  459 
=  709-3 ;  consequently,  by  dividing  the  greater  by  the  lesser,  and 


multiplying  by  the  given  volume,  we  get  *          X  1711  =  1808-66 

for  the  volume  at  the  temperature  of  250-3  degrees. 

Again,  if  the  elastic  force  at  the  lower  temperature  and  the  cor- 
responding volume  be  given,  the  elastic  force  at  the  higher  tem- 
perature can  readily  be  found ;  for  it  is  simply  as  the  volume  the 
vapour  occupies  at  the  lower  temperature  is  to  the  volume  at  the 
higher  temperature,  or  what  it  would  become  by  expansion,  so  is  the 
elastic  force  given  to  that  required. 

If  the  volume  which  steam  occupies  under  any  given  pressure 
and  temperature  be  given,  the  volume  which  it  will  occupy  under 
any  proposed  pressure  can  readily  be  found  by  reversing  the  pre- 
ceding process,  or  by  referring  to  chemical  tables  containing  the 


THE   STEAM   ENGINE.  201 

specific  gravity  of  the  gases  compared  with  air  as  unity  at  the  same 
pressure  and  temperature.  Now,  air  at  the  mean  state  of  the  at- 
mosphere has  a  specific  gravity  of  If  as  compared  with  water  at 
1000  ;  and  the  bulks  are  inversely  as  the  specific  gravities,  accord- 
ing to  the  general  laws  of  the  properties  of  matter  previously  an- 
nounced ;  hence  it  follows  that  air  is  818  times  the  bulk  of  an 
equal  weight  of  water,  for  1000  H-  If  =  818-18.  But,  by  the 
experiments  of  Dr.  Dalton,  it  has  been  found  that  steam  of  the 
same  pressure  and  temperature  has  a  specific  gravity  of  '625  com- 
pared with  air  as  unity  ;  consequently,  we  have  only  to  divide  the 
number  818-18  by  -625,  and  the  quotient  will  give  the  propor- 
tion of  volume  of  the  vapour  to  one  of  the  liquid  from  which  it  is 
generated  ;  thus  we  get  818-18  -5-  -625  =  1309  ;  that  is,  the  volume 
of  steam  at  60  degrees  of  Fahrenheit,  its  force  being  30  inches  of 
mercury,  is  1309  times  the  volume  of  an  equal  weight  of  water  ; 
hence  it  follows,  from  equation  (T),  that  when  the  temperature  in- 
creases to  t',  the  volume  becomes 

/459  +  t1  \ 
v'  =  1309  x  (459  +  6Q)  =  2-524(459  +  f); 

and  from  this  expression,  the  volume  corresponding  to  any  specified 
elastic  force  /,  and  temperature  t'9  may  easily  be  found  ;  for  it  is 
inversely  as  the  compressing  force  :  that  is, 

/:  30:  :  2-525(459  +  t')  :  vr  ; 
consequently,  by  working  out  the  analogy,  we  get 
=  75-67(459  +  f  ). 


By  this  theorem  is  found  the  volume  of  steam  as  compared  with 
that  of  the  water  producing  it,  when  under  a  pressure  correspond- 
ing to  the  temperature.  The  rule  in  words  is  as  follows  : 

RULE.  —  Calculate  the  elastic  force  in  inches  of  mercury  by  the 
rule  already  given  for  that  purpose,  and  reserve  it  for  a  divisor. 
To  the-  given  temperature  add  the  constant  number  459,  and  mul- 
tiply the  sum  by  75-67  ;  then  divide  the  product  by  the  reserved 
divisor,  and  the  quotient  will  give  the  volume  sought. 

When  the  temperature  of  steam  is  250-3  degrees  of  .Fahrenheit's 
thermometer,  what  is  the  volume,  compared  with  that  of  water  ? 

The  temperature  being  greater  than  212  degrees,  the  force  is  cal- 
culated by  the  rule  to  equation  (Q),  and  the  process  is  as  follows  : 

250-3  +  121  =  371-3  log.  2-5697249 
Constant  divisor  ==  333     log.  2-5224442  subtract 

(MJ472807  x6-42=0-3035421 
Atmosphere  =  30  inches  of  mercury          log.  1-4771213  add 

Elastic  force  =  60-348  log.  1*7806634  "| 

Again  it  is, 

459  +  250-3  =  709-3  log.  2-8508300)     ,,  fsub' 

Constant  coefficient  =  75-67  log.  1-8789237  j         4-7297537  J 

Volume  =  889-39  times  that  of  water,  log.  2^9490903  re 
mainder. 


202  THE    PRACTICAL    MODEL    CALCULATOR. 

Thus  we  have  given  the  method  of  calculating  the  elastic  force 
of  steam  when  the  temperature  is  given  either  in  atmospheres  or 
inches  of  mercury,  and  also  in  pounds  or  the  square  or  circular 
inch  :  we  have  also  reversed  the  process,  and  determined  the  tem- 
perature corresponding  '  to  any  given  elastic  force.  We  have, 
moreover,  shown  how  to  find  the  volume  corresponding  to  different 
temperatures,  when  the  pressure  is  constant  ;  and,  finally,  we  have 
calculated  the  volume,  when  under  a  pressure  due  to  the  elastic 
force.  These  are  the  chief  subjects  of  calculation  as  regards  the 
properties  of  steam  ;  and  we  earnestly  advise  our  readers  to  render 
themselves  familiar  with  the  several  operations.  The  calculations 
as  regards  the  motion  of  steam  in  the  parts  of  an  engine  to  produce 
power,  will  be  considered  in  another  part  of  the  present  treatise. 

The  equation  (U),  we  may  add,  can  be  exhibited  in  a  different 
form  involving  only  the  temperature  and  known  quantities  ;  for 
since  the  expressions  (P)  and  (Q)  represent  the  elastic  force  in  terms 
of  the  temperature,  according  as  it  is  under  or  above  212  degrees 
of  Fahrenheit,  we  have  only  to  substitute  those  values  of  the  elastic 
force  when  reduced  to  inches  of  mercury,  instead  of  the  symbol  / 
in  equation  (U),  and  we  obtain,  when  the  temperature  is  less  than 
212  degrees, 

Yol.=75-67(tem.+459)-^(-004016xtem.  +  -702807)7'71307.(V). 

and  when  the  temperature  exceeds  212  degrees,  the  expression  be- 
comes 


Vol.=75-67(tem.+459)H--005101xtem.  +  -617195f42  .  (W.) 

These  expressions  are  simple  in  their  form,  and  easily  reduced  ; 
but,  in  pursuance  of  the  plan  we  have  adopted,  it  becomes  necessary 
to  express  the  manner  of  their  reduction  in  words  at  length,  as 
follows  : 

RULE.  —  When  the  given  temperature  is  under  212  degrees,  mul- 
tiply the  temperature  in  degrees  of  Fahrenheit's  thermometer  by 
the  constant  fraction  -004016,  and  to  the  product  add  the  constant 
increment  -702807  ;  multiply  the  logarithm  of  the  sum  by  the  in- 
dex 7  '71307,  and  find  the  natural  or  common  number  answering  to 
the  product,  which  reserve  for  a  divisor.  To  the  temperature  add 
the  constant  number  459,  and  multiply  the  sum  by  the  coefficient 
75-67  for  a  dividend  ;  divide  the  latter  result  by  the  former,  and 
the  quotient  will  express  the  volume  of  steam  when  that  of  water  is 
unity. 

Again,  when  the  given  temperature  is  greater  than  212  degrees, 
multiply  it  by  the  fraction  -005101,  and  to  the  product  add  the 
constant  increment  -617195  ;  multiply  the  logarithm  of  the  sum 
by  the  index  6-42,  and  reserve  the  natural  number  answering  to 
the  product  for  a  divisor  ;  find  the  dividend  as  directed  above, 
which,  being  divided  by  the  divisor,  will  give  the  volume  of  steam 
when  that  of  the  water  is  unity. 

How  many  cubic  feet  of  steam  will  be  supplied  by  one  cubic  foot 


THE   STEAM   ENGINE.  203 

• 

of  water,  under  the  respective  temperatures  of  1ST  and  293-4  de- 
grees of  Fahrenheit's  thermometer  ? 
Here,  by  the  rule,  we  have 

187x0-004016=0-750992 
Constant  increment=0-702807 

Sum  =1-453799  log.  1625043  x  7'71307=1'2534069 
and  the  number  answering  to  this  logarithm  is  17-92284,  the  di- 
visor. But  187  +  459  =  646,  and  646  x  75-67  =  48882-82,  the 
dividend;  hence,  by  division,  we  get  48882-82 -r- 17-92284  = 
2727-4  cubic  feet  of  steam  from  one  cubic  foot  of  water. 
Again,  for  the  higher  temperature,  it  is 

293-4  x  0-005101  =  1-496633 
Constant  increment  =  0-617195 

Sum  =  2-113828 log.  0-3250696x642=2-0869468; 

and  the  number  answering  to  this  logarithm  is  122*165,  the  divisor. 
But  293-4  +  459  =  752-4,  and  752-4  x  75-67  =  56934-108,  the 
dividend ;  therefore,  by  division,  we  get  56934-108  -^  122-165  = 
466-04  cubic  feet  of  steam  from  one  cubic  foot  of  water. 

The  preceding  is  a  very  simple  process  for  calculating  the  volume 
which  the  steam  of  a  cubic  foot  of  water  will  occupy  when  under 
a  pressure  due  to  a  given  temperature  and  elastic  force  ;  and  since 
a  knowledge  of  this  particular  is  of  the  utmost  importance  in  cal- 
culations connected  with  the  steam  engine,  it  is  presumed  that  our 
readers  will  find  it  to  their  advantage  to  render  themselves  familiar 
with  the  method  of  obtaining  it.  The  above  example  includes  both 
cases  of  the  problem,  a  circumstance  which  gives  to  the  operation, 
considered  as  a  whole,  a  somewhat  formidable  appearance :  but  it 
would  be  difficult  to  conceive  a  case  in  actual  practice  where  the 
application  of  both  the  formulas  will  be  required  at  one  and  the 
same  time ;  the  entire  process  must  therefore  be  considered  as  em- 
bracing only  one  of  the  cases  above  exemplified  ;  and  consequently 
it  can  be  performed  with  the  greatest  facility  by  every  person  who 
is  acquainted  with  the  use  of  logarithms ;  and  those  unacquainted 
with  the  application  of  logarithms  ought  to  make  themselves  masters 
of  that  very  simple  mode  of  computation. 

Another  thing  which  it  is  necessary  sometimes  to  discover  in 
reasoning  on  the  properties  of  steam  as  referred  to  its  action  in  a 
steam  engine,  is  the  weight  of  a  cubic  foot,  or  any  other  quantity 
of  it,  expressed  in  grains,  corresponding  to  a  given  temperature  and 
pressure.  Now,  it  has  been  ascertained  by  experiment,  that  when 
the  temperature  of  steam  is  60  degrees  of  Fahrenheit,  and  the 
pressure  equal  to  30  inches  of  mercury,  the  weight  of  a  cubic  foot 
in  grains  is  329-4;  but  the  weight  is  directly  proportional  to  th<* 
elastic  force,  for  the  elastic  force  is  proportional  to  the  density  : 
consequently,  if/  denote  any  other  elastic  force,  and  w  the  weight 
in  grains  corresponding  thereto,  then  we  have 

30  :/::  329-4  :  w  =  10-98 /, 


204  THE   PRACTICAL   MODEL   CALCULATOR. 

the  weight  of  a  cubic  foot  of  vapour  at  the  force/,  and  temperature 
60  decrees  of  Fahrenheit.  Let  t  denote  the  temperature  at  the 

459  +  t       459  +  t 
force/;  then  by  equation  (1),  we  have  v  =  fgcfqrgo  =  ~5T9~> 

the  volume  at  the  temperature  t,  supposing  the  volume  at  60  de- 
grees to  be  unity  ;  that  is,  one  cubic  foot.  Now,  since  the  den- 
sities are  inversely  proportional  to  the  spaces  which  the  vapour  oc- 


(459  +  t)  , 

cupies,  we  have     ~  gjg  —  '  '  1  •  •  ™  '  w  =  459  +  t  ;  but  by  the 


preceding  analogy,  the  value  of  w  is  10  -98/;  therefore,  by  substi- 
tution, we  get 

569S-62/ 
•  •    •    •    ' 


This  equation  expresses  the  weight  in  grains  of  a  cubic  foot  of 
steam  at  the  temperature  t  and  force/;  and  if  we  substitute  the 
value  of  /,  from  equations  (P)  and  (Q),  reduced  to  inches  of  mer- 
cury, and  modified  for  the  two  cases  of  temperature  below  and 
above  212  degrees  of  Fahrenheit,  we  shall  obtain,  in  the  first  case, 

w'  =  (0-012324  x  temp.  +  2-155611)7'71307  -r-  (temp.  +  459).  .  .  .  (Y) 
and  for  the  second  case,  where  the  temperature  exceeds  212,  it  is 
wf  =  (0-01962  x  temp.  +  2-373T4)6'42  -r-  (temp.  +  459)  .  .  .  (Z) 

These  two  equations,  like  those  marked  (V)  and  (W)  are  suf- 
ficiently simple  in  their  form,  and  oifer  but  little  difficulty  in  their 
application.  The  rule  for  their  reduction  when  expressed  in  words 
at  length,  is  as  follows  : 

RULE.  —  When  the  temperature  is  less  than  212  degrees,  multi- 
ply the  given  temperature,  in  degrees  of  Fahrenheit's  thermometer, 
by  the  fraction  0*012324,  and  to  the  product  add  the  constant  in- 
crement 2-155611  ;  then  multiply  the  logarithm  of  the  sum  by  the 
index  7*71307,  and  from  the  product  subtract  the  logarithm  of  the 
temperature,  increased  by  459  ;  the  natural  number  answering  to 
the  remainder  will  be  the  weight  of  a  cubic  foot  in  grains. 

Again,  when  the  temperature  exceeds  212,  multiply  it  by  the 
fraction  0-01962,  and  to  the  product  add  the  constant  increment 
2-37374  ;  then  multiply  the  logarithm  of  the  sum  by  the  index  6-42, 
and  from  the  product  subtract  the  logarithm  of  the  temperature  in- 
creased by  459  ;  the  natural  number  answering  to  the  remainder 
will  be  the  weight  of  a  cubic  foot  in  grains. 

Supposing  the  temperatures  to  be  as  in  the  preceding  example, 
what  will  be  the  weight  of  a  cubic  foot  in  grains  for  the  two  cases  ? 

Here,  by  the  rule,  we  have 

187  X  0-012324  ==  2-304588 
Constant  increment  =  2-155611 

Sum  =  4-460199   log.  0-6493542  X  7-71307  =  5-0085143 
187  -f  459  ^  646         .....        log.  2-8102325,  subtract 

Natural  number  =  157-863  grains  per  cubic  foot         log.  2-1982818 


THE    STEAM    ENGINE.  205 

For  the  higher  temperature,  it  is 

293-4  x  0-01962  =  5-756508 
Constant  increment  =  2-378740 

Sum  =  8-130248      log.  0-9101038  X  6-42  =  5-8428664 
293-4  -f  459  =  752-4         .        .        .        .        log.  2-8764488,  subtract 

Natural  number  =  925-59  grains  per  cubic  foot  .  log.  2-9664176 
Here  again  the  operation  resolves  both  cases  of  the  problem ; 
but  in  practice  only  one  of  them  can  be  required. 

THE    MOTION   OF   ELASTIC   FLUIDS. 

The  next  subject  that  claims  our  attention  is  the  velocity  with 
which  elastic  fluids  or  vapours  move  in  pipes  or  confined  passages. 
It  is  a  well-known  fact  in  the  doctrine  of  pneumatics,  that  the  mo- 
tion of  free  elastic  fluids  depends  upon  the  temperature  and  pres- 
sure of  the  atmosphere ;  and,  consequently,  when  an  elastic  fluid 
is  confined  in  a  close  vessel,  it  must  be  similarly  circumstanced 
with  regard  to  temperature  and  pressure  as  it  would  be  in  an  at- 
mosphere competent  to  exert  the  same  pressure  upon  it.  The  sim- 
plest and  most  convenient  way  of  estimating  the  motion  of  an  elastic 
fluid  is  to  assign  the  height  of  a  column  of  uniform  density,  capable 
of  producing  the  same  pressure  as  that  which  the  fluid  sustains  in 
its  state  of  confinement ;  for  under  the  pressure  of  such  a  column, 
the  velocity  into  a  perfect  vacuum  will  be  the  same  as  that  acquired 
by  a  heavy  body  in  falling  through  the  height  of  the  homogeneous 
column,  a  proper  allowance  being  made  for  the  contraction  at  the 
aperture  or  orifice  through  which  the  fluid  flows. 

When  a  passage  is  opened  between  two  vessels  containing  fluids 
of  different  densities,  the  fluid  of  greatest  density  rushes  out  of  the 
vessel  that  contains  it,  into  th&  one  containing  the  rarer  fluid,  and 
the  velocity  of  influx  at  the  first  instant  of  the  motion  is  equal  to 
that  which  a  heavy  body  acquires  in  falling  through  a  certain 
height,  and  that  height  is  equal  to  the  difference  of  two  uniform 
columns  of  the  fluid  of  greatest  density,  competent  to  produce  the 
pressures  under  which  the  fluids  are  originally  confined  ;  and  the 
velocity  of  motion  at  any  other  instant  is  proportional  to  the  square 
root  of  the  difference  between  the  heights  of  the  uniform  columns 
producing  the  pressures  at  that  instant.  Hence  we  infer  that  the 
velocity  of  motion  continually  decreases,— the  density  of  the  fluids 
in  the  two  vessels  approaching  nearer  and  nearer  to  an  equality, 
and  after  a  certain  time  an  equilibrium  obtains,  and  the  velocity 
of  motion  ceases. 

It  is  abundantly  confirmed  by  observation  and  experiment,  that 
oblique  action  produces  very  nearly  the  same  effect  in  the  motion 
of  elastic  fluids  through  apertures  as  it  does  in  the  case  of  water ; 
and  it  has  moreover  been  ascertained  that  eddies  take  place  under 
similar  circumstances,  and  these  eddies  must  of  course  have  a  ten- 
dency to  retard  the  motion :  it  therefore  becomes  necessary,  in  all 
the  calculations  of  practice,  to  make  some  allowance  for  the  retard- 
ation that  takes  place  in  passing  the  orifice ;  and  this  end  is  most 


206  THE   PRACTICAL   MODEL    CALCULATOR. 

conveniently  answered  by  modifying  the  constant  coefficient  ac- 
cording to  the  nature  of  the  aperture  through  which  the  motion 
is  made.  Numerous  experiments  have  been  made  to  ascertain  the 
effect  of  contraction  in  orifices  of  different  forms  and  under  dif- 
ferent conditions,  and  amongst  those  which  have  proved  the  most 
successful  in  this  respect,  we  may  mention  the  experiments  of  Du 
Buat  and  Eytelwein,  the  latter  of  whom  has  supplied  us  with  a 
series  of  coefficients,  which,  although  not  exclusively  applicable  to 
the  case  of  the  steam  engine,  yet,  on  account  of  their  extensive 
utility,  we  take  the  liberty  to  transcribe.  They  are  as  follow : — 

1.  For  the  velocity  of  motion  that  would  re- 
sult from  the  direct  unretarded  action  of 

the  column  of  the  fluid  that  produces  it,  we  

have 3V  =  x/579A 

2.  For  an  orifice  or  tube  in  the  form  of  the 

contracted  vein 10  V  =  v"6084A 

3.  For  wide  openings  having  the  sill  on  a> 
level  with  the  bottom  of  the  reservoir  ... 

4.  For  sluices  with  walls  in  a  line  with  the  J>10  V  =  \/5929A 
orifice 

5.  For  bridges  with  pointed  piers 

6.  For  narrow  openings  having  the  sill  on  a 
level  with  the  bottom  of  the  reservoir  ... 

7.  For  small  openings  in  a  sluice  with  side 
walls 

8.  For  abrupt  projections 

9.  For  bridges  with  square  piers.. 

10.  For  openings  in  sluices  without  side  walls  10  V  =  v/26017i 

11.  For  openings  or  orifices  in  a  thin  plate V  =  \/257i 

12.  For  a  straight  tube  from  2  to  3  diameters 

in  length  projecting  outwards 10  V  =  -v/4225 

13.  For  a  tube  from  2  to  3  diameters  in  length 


projecting  inwards 10  V  =  N/2976'25^ 

It  is  necessary  to  observe,  that  in  all  fliese  equations  V  is  the 
velocity  of  motion  in  feet  per  second,  and  7i  the  height  of  the  co- 
lumn producing  it,  estimated  also  in  feet.  Nos.  1,  2,  11,  12,  and 
13  are  those  which  more  particularly  apply  to  the  usual  passages 
for  the  steam  in  a  steam  engine  ;  but  since  all  the  others  meet  their 
application  in  the  every-day  practice  of  the  civil  engineer,  we  have 
thought  it  useful  to  supply  them. 


MOTION   OF   STEAM   IN   AN   ENGINE. 


We  have  already  stated  that  the  best  method  of  estimating  the 
motion  of  an  elastic  fluid!',  such  as  steam  or  the  vapour  of  water,  is 
to  assign  the  height  of  a  uniform  column  of  that  fluid  capable  of 
producing  the  pressure  :  the  determination  of  this  column  is  there- 
fore the  leading  step  of  the  inquiry ;  and  since  the  elastic  force  of 
steam  is  usually  reckoned  in  inches  of  mercury,  30  inches  being 


THE   STEAM   ENGINE.  207 

equal  to  the  pressure  of  the  atmosphere,  the  subject  presents  but 
little  difficulty ;  for  we  have  already  seen  'that  the  height  of  a  co- 
lumn of  water  of  the  temperature  of  60  degrees,  balancing  a  column 
of  80  inches  of  mercury,  is  34-023  feet ;  the  corresponding  column 
of  steam  must  therefore  be  as  its  relative  bulk  and  elastic  force ; 
hence  we  have  30  :  34-023  :fv:h  =  1-1341  fv,  where  /  is  the 
elastic  force  of  the  steam  in  inches  of  mercury,  v  the  correspond- 
ing volume  or  bulk  when  that  of  water  is  unity,  and  h  the  height 
of  a  uniform  column  of  the  fluid  capable  of  producing  the  pressure 
due  to  the  elastic  force ;  consequently,  in  the  case  of  a  direct  un- 
retarded  action,  the  velocity  into  a  perfect  vacuum,  according  to 
No.  1  of  the  preceding  class  of  formulas,  is  V  =  8-542  \/fv ;  but 
for  the  best  form  of  pipes,  or  a  conical  tube  in  form  of  the  con- 
tracted vein,  the  velocity  into  a  vacuum,  according  to  No.  2,  be- 
comes V  =  8-307  */f  v\  and  for  pipes  of  the  usual  construction, 
No.  12  gives  V  =  6-922  v'/v ;  No.  13  gives  V  =  5-804  x//V; 
and  in  the  case  of  a  simple  orifice  in  a  thin  plate,  we  get  from 
No.  11  V  =  5-322  -v/f  v.  The  consideration  of  all  these  equa- 
tions may  occasionally  be  required,  but  our  researches  will  at  pre- 
sent be  limited  to  that  arising  from  No.  12,  as  being  the  best 
adapted  for  general  practice ;  and  for  the  purpose  of  shortening 
the  investigation,  we  shall  take  no  further  notice  of  the  case  in 
which  the  temperature  of  the  steam  is  below  212  degrees  of  Fah- 
renheit ;  for  the  expression  which  indicates  the  velocity  into  a  va- 
cuum being  independent  of  the  elastic  force,  a  separate  considera- 
tion for  the  two  cases  is  here  unnecessary. 

It  has  been  shown  in  the  equation  marked  (U),  that  the  volume 
of  steam  which  is  generated  from  an  unit  of  water,  is  v  = 
75-67  (temp.  +  459) 

— -T, —  — ;  let  this  value  of  v  be  substituted  for  it  in 

the  equation  V  =  6-922  Vf  v,  and  we  obtain  for  the  velocity  into 
a  vacuum  for  the  usual  form  of  steam  passages,  as  follows,  viz.  : 

V  =  60-2143  N/(temp.  H-  459). 

This  is  a  very  neat  and  simple  expression,  and  the  object  de- 
termined by  it  is  a  very  important  one :  it  therefore  merits  the 
reader's  utmost  attention,  especially  if  he  is  desirous  of  becoming 
familiar  with  the  calculations  in  reference  to  the  motion  of  steam. 
The  rule  which  the  equation  supplies,  when  expressed  in  words  at 
length,  is  as  follows  : — 

RULE. — To  the  temperature  of  the  steam,  in  degrees  of  Fahren- 
heit's thermometer,  add  the  constant  number  or  increment  459,  and 
multiply  the  square  root  of  the  sum  by  60-2143 ;  the  product  will 
be  the  velocity  with  which  the  steam  rushes  into  a  vacuum  in  feet 
per  second. 

With  what  velocity  will  steam  of  293-4  degrees  of  Fahrenheit's 
thermometer  rush  into  a  vacuum  when  under  a  pressure  due  to  the 
elastic  force  corresponding  to  the  given  temperature. 


208  THE   PRACTICAL   MODEL   CALCULATOR. 

By  the  rule  it  is  293-4  -f  459  =     752-4 £  log.  1-4382244 

Constant  coefficient  =  60-2143 log.  1-7797018  add 

Velocity  into  a  vacuum  in  feet  per  second  =  1651-68 log.  3-21792G2 

This  is  the  velocity  into  a  perfect  vacuum,  when  the  motion  is 
made  through  a  straight  pipe  of  uniform  diameter ;  but  when  the 
pipe  is  alternately  enlarged  and  contracted,  the  velocity  must  ne- 
cessarily be  reduced  in  proportion  to  the  nature  of  the  contraction ; 
and  it  is  further  manifest,  that  every  bend  and  angle  in  a  pipe  will 
be  attended  with  a  correspondent  diminution  in  the  velocity  of  mo- 
tion :  it  therefore  behoves  us,  in  the  actual  construction  of  steam 
passages,  to  avoid  these  causes  of  loss  as  much  as  possible ;  and 
where  they  cannot  be  avoided  altogether,  such  forms  should  be 
adopted  as  will  produce  the  smallest  possible  retarding  effect.  In 
cases  where  the  forms  are  limited  by  the  situation  and  conditions 
of  construction,  such  corrections  should  be  applied  as  the  circum- 
stances of  the  case  demand ;  and  the  amount  of  these  corrections 
must  be  estimated  according  to  the  nature  of  the  obstructions  them- 
selves. For  each  right-angled  bend,  the  diminution  of  velocity  is 
usually  set  down  as  being  about  one-tenth  of  its  unobstructed  value  ; 
but  whether  this  conclusion  be  correct  or  not,  it  is  at  least  certain 
that  the  obstruction  .  in  the  case  of  a  right-angled  bend  is  much 
greater  than  in  that  of  a  gradually  curved  one.  It  is  a  very  com- 
mon thing,  especially  in  steam  vessels,  for  the  main  steam  pipe  to 
send  off  branches  at  right  angles  to  each  cylinder,  and  it  is  easy  to 
see  that  a  great  diminution  in  the  velocity  of  the  steam  must  take 
place  here.  In  the  expansion  valve  chest  a  further  obstruction 
must  be  met  with,  probably  to  the  extent  of  reducing  the  velocity 
of  the  steam  two-tenths  of  its  whole  amount. 

These  proportional  corrections  are  not  to  be  taken  as  the  results 
of  experiments  that  have  been  performed  for  the  purpose  of  deter- 
mining the  effect  of  "the  above  causes  of  retardation:  we  have  no 
experiments  of  this  sort  on  which  reliance  can  be  placed ;  and,  in 
consequence,  such  elements  can  only  be  inferred  from  a  comparison 
of  the  principles  that  regulate  the  motion  of  other  fluids  under  simi- 
lar circumstances :  they  will,  however,  greatly  assist  the  engineer 
in  arriving  at  an  approximate  estimate  of  the  diminution  that  takes 
place  in  the  velocity  in  passing  any  number  of  obstructions,  when 
the  precise  nature  of  those  obstructions  can  be  ascertained.  In  the 
generality  of  practical  cases,  if  the  constant  coefficient  60*2143  be 
reduced  in  the  ratio  of  650  to  450,  the  resulting  constant  41-6868 
may  be  employed  without  introducing  an  error  of  any  consequence. 

OF   THE  ASCENT   OF    SMOKE   AND    HEATED    AIR   IN   CHIMNEYS. 

The  subject  of  chimney  flues,  with  the  ascent  of  smoke  and  heated 
air,  is  another  case  of  the  motion  of  elastic  fluids,  in  which,  by  a 
change  of  temperature,  an  atmospheric  column  assumes  a  different 
density  from  another,  where  no  such  alteration  of  temperature  oc- 
curs. ^  The  proper  construction  of  chimneys  is  a  matter  of  very 
great  importance  to  the  practical  engineer,  for  in  a  close  fireplace, 


THE    STEAM    ENGINE.  209 

designed  for  the  generation  of  steam,  there  must  be  a  considerable 
draught  to  accomplish  the  intended  purpose,  and  this  depends  upon 
the  three  following  particulars,  viz.  : 

1.  The  height  of  the  chimney  from  the  throat  to  the  top. 

2.  The  area  of  the  transverse  section. 

3.  The  temperature  at  which  the  smoke  and  heated  air  are  al- 
lowed to  enter  it. 

The  formula  for  determining  the  power  of  the  chimney  may  be 
investigated  in  the  following  manner  : 

Put  h  =  the  height  in  feet  from  the  place  where  the  flue  enters 

to  the  top  of  the  chimney, 
b  =  the  number  of  cubic  feet  of  air  of  atmospheric  density 

that  the  chimney  must  discharge  per  hour, 
a  —  the  area  of  the  aperture  in  square  inches  through  which 

b  cubic  feet  of  air  must  pass  when  expanded  by  a 

change  of  temperature, 

v  =  the  velocity  of  ascent  in  feet  per  second, 
t1  =  the  temperature  of  the  external  air,  and 
t  —  the  temperature  of  the  air  to  be  discharged  by  the 

chimney. 

Now  the  force  producing  the  motion  in  this  case  is  manifestly 
the  difference  between  the  weight  of  a  column  of  the  atmospheric 
air  and  another  of  the  air  discharged  by  the  chimney  :  and  when 
the  temperature  of  the  atmospheric  air  is  at  52  degrees  of  Fahren- 
heit's thermometer,  this  difference  will  be  indicated  by  the  term 

/*  \p-3I~459  )  ;  the  velocity  of  ascent  will  therefore  be 

v  —  J  64f  h  |  p~^T459  f  feet  Per  second,  and  the  quantity  of  air 


discharged  per  second  will  therefore  be,  a  J  64  f  )  I  ? 

supposing  that  there  is  no  contraction  in  the  stream  of  air  ;  but  it 
is  found  by  experiment,  that  in  all  cases  the  contraction  that  takes 
place  diminishes  the  quantity  discharged,  by  about  three-eighths  of 
the  whole  ;  consequently,  the  quantity  discharged  per  hour  in  cu- 
bic feet  becomes 


This  would  be  the  quantity  discharged,  provided  there  were  no 
increase  of  volume  in  consequence  of  the  change  of  temperature  ; 

but  air  expands  from  b  to     ^    .    459     for  *'  —  *  degrees  of  tem- 
s been  show 

b  (if  -f  459) 


perature,  as  has  been  shown  elsewhere  ;  consequently,  by  compa- 
rison, we  have 


125  b9        t  +  459  ' 
s  2  H 


210  THE    PRACTICAL   MODEL   CALCULATOR. 

From  this  equation,  therefore,  any  one  of  the  quantities  which 
it  involves  can  be  found,  when  the  others  are  given  :  it  however 
supposes  that  there  is  no  other  cause  of  diminution  but  the  contrac- 
tion at  the  aperture  ;  but  this  can  seldom  if  ever  be  the  case  ;  for 
eddies,  loss  of  heat,  obstructions,  and  change  of  direction  in  the 
chimney,  will  diminish  the  velocity,  and  consequently  a  larger  area 
will  be  required  to  suifer  the  heated  air  to  pass.  A  sufficient  al- 
lowance for  these  causes  of  retardation  will  be  made,  if  we  change 
the  coefficient  125*69  to  100  ;  and  in  this  case  the  equation  for  the 
area  of  section  becomes 


459)3  -r-  100  (t  +  459)  </h  (tf  -  *). 

And  if  we  take  the  mean  temperature  of  the  air  of  the  atmo- 
sphere at  52  degrees  of  Fahrenheit,  and  make  an  allowance  of  16 
degrees  for  the  difference  of  density  between  atmospheric  air  and 
coal  smoke,  our  equation  will  ultimately  assume  the  form  . 


a  =  b  V(t'  +  459)3  -T-  51100  Vh  (f  -  *  -  16). 

It  has  been  found  by  experiment  that  200  cubic  feet  of  air  of  at- 
mospheric density  are  required  for  the  complete  combustion  of  one 
pound  of  coal,  and  the  consumption  of  ten  pounds  of  coal  per  hour 
is  usually  reckoned  equivalent  to  one  horse  power :  it  therefore  ap- 
pears that  2000  cubic  feet  of  air  per  hour  must  pass  through  the  fire 
for  each  horse  power  of  the  engine.  This  is  a  large  allowance,  but 
it  is  the  safest  plan  to  calculate  in  excess  in  the  first  instance  ;  for 
the  chimney  may  afterwards  be  convenient,  even  if  considerably 
larger  than  is  necessary.  The  rule  for  reducing  the  equation  is  as 
follows : — 

RULE. — Multiply  the  number  of  horse  power  of  the  engine  by 
the  |  power  of  the  temperature  at  which  the  air  enters  the  chimney, 
increased  by  459;  then  divide  the  product  by  25-55  times  the 
square  root  of  the  height  of  the  chimney  in  feet,  multiplied  by  the 
difference  of  temperature,  less  16  degrees,  and  the  quotient  will  be 
the  area  of  the  chimney  in  square  inches. 

Suppose  the  height  of  the  chimney  for  a  40-horse  engine  to  be 
70  feet,  what  should  be  its  area  when  the  difference  between  the 
temperature  at  which  the  air  enters  the  flue,  and  that  of  the 
atmosphere  is  250  degrees  ? 

Here,  by  the  rule,  we  have, 

250  -f  52  =  302,  the  temperature  at  which  the  air  enters 
Constant  increment  =  459  [the  flue. 

Sum  =  761 log.  2-8813847 

3 

2)8-6441541 

4-3220770 
Number  of  horse  power  =  40 log.  1-6020600 

5-9241370 


THE   STEAM   ENGINE.  211 

5-9241370 

250  -  16  ==  234  ....  log.  2-3692159 
height  =  70  feet   .  .  log.  1-8450980 

2)4-2143139 

2-1071569  \ 
Constant  =  25-55  .     .     log.  1.4073909  /     .     .     .   3-5145478 

Hence  the  area  of  the  chimney  in  square  inches  is  256-79,  log. 
2-4095892;  and  in  this  way  may  the  area  be  calculated  for  any 
other  case ;  but  particular  care  must  be  taken  to  have  the  data  ac- 
curately determined  before  the  calculation  is  begun.  In  the  above 
example  the  particulars  are  merely  assumed ;  but  even  that  is  suffi- 
cient to  show  the  process  of  calculation,  which  is  more  immediately 
the  object  of  the  present  inquiry.  It  is  right,  however,  to  add, 
that  recent  experiments  have  greatly  shaken  the  doctrine  that  it  is 
beneficial  to  make  chimneys  small  at  the  top,  though  such  is  the 
way  in  which  they  are,  nevertheless,  still  constructed,  and  our  rules 
must  have  reference  to  the  present  practice.  It  appears,  however, 
that  it  would  be  the  best  way  to  make  chimneys  expand  as  they 
ascend,  after  the  manner  of  a  trumpet,  with  its  mouth  turned  down- 
wards :  but  these  experiments  require  further  confirmation. 

The  method  of  calculation  adopted  above  is  founded  on  the  prin- 
ciple of  correcting  the  temperature  for  the  difference  between  the 
specific  gravity  of  atmospheric  air  and  that  of  coal-smoke,  the  one 
being  unity  and  the  other  1-05  ;  there  is,  however,  another  method, 
somewhat  more  elegant  and  legitimate,  by  employing  the  specific 
gravity  of  coal-smoke  itself:  the  investigation  is  rather  tedious  and 
prolix,  but  the  resulting  formula  is  by  no  means  difficult ;  and  since 
both  methods  give  the  same  result  when  properly  calculated,  we 
make  no  further  apology  for  presenting  our  readers  with  another 
rule  for  obtaining  the  same  object.  The  formula  is  as  follows : 

=  b(t'  +  459)   /~  JT 

2757-5     \ h  (£'-77-55) 

where  a  is  the  area  of  the  transverse  section  of  the  chimney  in 
square  inches,  b  the  quantity  of  atmospheric  air  required  for  com- 
bustion of  the  coal  in  cubic  feet  per  hour,  h  the  height  of  the  chim- 
ney in  feet,  and  tf  the  temperature  at  which  the  air  enters  the  flue 
after  passing  through  the  fire.  The  rule  for  performing  this  pro- 
cess is  thus  expressed : 

RULE. — From  the  temperature  at  which  the  air  enters  the  chim- 
ney, subtract  the  constant  decrement  77-55  ;  multiply  the  remainder 
by  the  height  of  the  chimney  in  feet,  divide  unity  by  the  product, 
and  extract  the  square  root  of  the  quotient.  To  the  temperature 
of  the  heated  air,  add  the  constant  number  459 ;  multiply  the  sum 
by  the  number  of  cubic  feet  required  for  combustion  per  hour,  and 
divide  the  product  by  the  number  2757 '5 ;  then  multiply  the  quo- 
tient by  the  square  root  found  as  above,  and  the  product  will  be  the 
number  of  square  inches  in  the  transverse  section  of  the  chimney. 


212  THE    PRACTICAL   MODEL    CALCULATOR. 

Suppose  a  mass  of  fuel  in  a  state  of  combustion  to  require  5000 
cubic  feet  of  air  per  hour,  what. must  be  the  size  of  the  chimney 
when  its  height  is  100  feet,  the  temperature  at  which  the  heated 
air  enters  the  chimney  being  200  degrees  of  Fahrenheit's  ther- 
mometer ? 

By  the  rule  we  have  200-77-55=122-45  .  .  log.  2-0879588 
Height  of  the  chimney =100.  .  .  .  log.  2-0000000 

4-0879588 
2)5-9120412 

7-9560206 
200+459=659  .  .  .  log.  2-8188854)  , 

5000  .  .  .  log.  3-6989700  V  add  3-0773399 

2757-5  ar.  co.  log.  6-5594845  J         

1-0333605     10-798  in. 

This  appears  to  be  a  very  small  flue  for  the  quantity  of  air  that 
passes  through  it  per  hour;  but  it  must  be  observed  that  wre  have 
assumed  a  great  height  for  the  shaft,  which  has  the  effect  of  cre- 
ating a  very  powerful  draught,  thereby  drawing  off  the  heated  air 
with  great  rapidity. 

The  advantage  of  a  high  flue  is  so  very  great,  that  the  reader 
may  be  desirous  of  knowing  to  what  height  a  chimney  of  a  given 
base  may  be  carried  with  safety,  in  cases  where  it  is  inconvenient 
to  secure  it  with  lateral  stays ;  and,  as  an  approximate  rule  for  this 
purpose  is  not  difficult  of  investigation,  we  think  proper  to  supply 
it  here. 

When  the  chimney  is  equally  wide  throughout  its  whole  height, 
the  formula  is 

156 


12000-  J7iw; 

but  when  the  side  of  the  base  is  double  the  size  of  the  top,  the 
equation  becomes 


/ 
\ 


104 


12000_0-42Aw; 

where  s  is  the  side  of  the  base  in  feet,  h  the  height,  and  m  the 
weight  of  one  cubic  foot  of  the  material.  When  the  chimney  stalk 
is  not  square,  but  longer  on  the  one  side  than  the  other,  s  must  be 
the  least  dimension.  The  proportion  of  solid  wall  to  a  given  base, 
as  sanctioned  by  experience,  is  about  two-thirds  of  its  area,  conse- 
quently w  ought  to  be  two-thirds  of  the  weight  of  a  cubic  foot  of 
brickwork.  Now,  a  cubic  foot  of  dried  brickwork  is,  on  an  average, 
114  Ibs.  ;  consequently  w  —  76  Ibs.  ;  and  if  this  be  substituted  in 
the  foregoing  equations,  we  get  for  a  chimney  of  equal  size  through- 
out, 


/ 156 

\1200  -  i 


156 
25  /*; 


THE   STEAM   ENGINE.  213 

and  when  the  chimney  tapers  to  one-half  the  size  at  top,  it  is 

104 


12000  -  32  h  ; 

where  it  may  be  remarked  that  12000  Ibs.  is  the  cohesive  force  of 
one  square  foot  of  mortar  ;  and  in  the  investigation  of  the  formulas 
we  have  assumed  the  greatest  force  of  the  wind  on  a  square  foot 
of  surface  at  52  Ibs.  These  equations  are  too  simple  in  their  form 
to  require  elucidation  from  us;  we  therefore  leave  the  reduction  as 
an  exercise  to  the  reader,  who  it  is  presumed  will  find  no  difficulty 
in  resolving  the  several  cases  that  may  arise  in  the  course  of  his 
practice. 

is  the  expression  given  by  M.  Pdclet  for  the  velocity  of  smoke  in 
a  chimney.  v9  the  velocity  ;  £,  the  temperature,  whose  maximum 
value  is  about  300°  centigrade  ;  g  =  32^  feet  ;  D,  the  diameter 
of  the  chimney  ;  H,  the  height  ;  L,  the  length  of  horizontal  flues. 
supposing  them  formed  into  a  cylinder  of  the  same  diameter 
as  that  of  the  chimney.  K  =  -0127  for  brick,  =  '005  for  sheet- 
iron,  and  =  -0025  for  cast-iron  chimneys,  a  =  -00365. 
Let  L=60;  H=150;  D=5  ;  K^='005;  2#=64J;  *=300°; 


/      ZqRatD 
^=•00365.     Then  v=  \D+2<yK(H+L)  =  26'986  feet' 

A  cubic  foot  of  water  raised  into  steam  is  reckoned  equivalent  to 
a  horse  power,  and  to  generate  the  steam  with  sufficient  rapidity, 
an  allowance  of  one  square  foot  of  fire-bars,  and  one  square  yard 
of  effective  heating  surface,  are  very  commonly  made  in  practice, 
at  least  in  land  engines.  These  proportions,  however,  greatly  vary 
in  different  cases  ;  and  in  some  of  the  best  marine  engine  boilers, 
where  the  area  of  fire-grate  is  restricted  by  the  breadth  of  the  ves- 
sel, and  the  impossibility  of  firing  long  furnaces  effectually  at  sea, 
half  a  square  foot  of  fire-grate  per  horse  power  is  a  very  common 
proportion.  Ten  cubic  feet  of  water  in  the  boiler  per  horse  power, 
and  ten  cubic  feet  of  steam  room  per  horse  power,  have  been  as- 
signed as  the  average  proportion  of  these  elements  ;  but  the  fact  is, 
no  general  rule  can  be  formed  upon  the  subject,  for  the  proportions 
which  would  be  suitable  for  a  wagon  boiler  would  be  inapplicable 
to  a  tubular  boiler,  whether  marine  or  locomotive  ;  and  good  ex- 
amples will  in  such  cases  be  found  a  safer  guide  than  rules  which 
must  often  give  a  false  result.  A  capacity  of  three  cubic  feet  per 
horse  power  is  a  common  enough  proportion  of  furnace-room,  and 
it  is  a  good  plan  to  make  the  furnaces  of  a  considerable  width,  as 
they  can  then  be  fired  more  effectually,  and  do  not  produce  so  much 
smoke  as  if  they  are  made  narrow.  As  regards  the  question  of 
draft,  there  is  a  great  difference  of  opinion  among  engineers  upon 
the  subject,  some  preferring  a  very  slow  draft  and  others  a  rapid 
one.  It  is  obvious  that  the  question  of  draft  is  virtually  that  of 


214  THE    PRACTICAL   MODEL   CALCULATOR. 

the  area  of  fire-grate,  or  of  the  quantity  of  fuel  consumed  upon  a 
given  area  of  grate  surface,  and  the  weight  of  fuel  burned  on  a  foot 
of  fire-grate  per  hour  varies  in  different  cases  in  practice  from  3J 
to  80  Ibs.  Upon  the  quickness  of  the  draft  again  hinges  the  ques- 
tion of  the  proper  thickness  of  the  stratum  of  incandescent  fuel 
upon  the  grate ;  for  if  the  draft  be  very  strong,  and  the  fire  at  the 
same  time  be  thin,  a  great  deal  of  uncombined  oxygen  will  escape 
up  through  the  fire,  and  a  needless  refrigeration  of  the  contents  of 
the  flues  will  be  thereby  occasioned ;  whereas,  if  the  fire  be  thick, 
and  the  draft  be  sluggish,  much  of  the  useful  effect  of  the  coal  will 
be  lost  by  the  formation  of  carbonic  oxide.  The  length  of  the  cir- 
cuit made  by  the  smoke  varies  in  almost  every  boiler,  and  the  same 
may  be  said  of  the  area  of  the  flue  in  its  cross  section,  through 
which  the  smoke  has  to  pass.  As  an  average,  about  one-fifth  of 
the  area  of  fire-grate  for  the  area  of  the  flue  behind  the  bridge, 
diminished  to  half  that  amount  for  the  area  of  the-  chimney,  has 
been  given  as  a  good  proportion,  but  the  examples  which  we  have 
given,  and  the  average  flue  area  of  the  boilers  which  we  shall 
describe,  may  be  taken  as  a  safer  guide  than  any  such  loose  state- 
ments. When  the  flue  is  too  long,  or  its  sectional  area  is  insuffi- 
cient, the  draft  becomes  insufficient  to  furnish  the  requisite  quantity 
of  steam ;  whereas  if  the  flue  be  too  short  or  too  large  in  its  area, 
a  large  quantity  of  the  heat  escapes  up  the  chimney,  and  a  depo- 
sition of  soot  in  the  flues  also  takes  place.  This  last  fault  is  one 
of  material  consequence  in  the  case  of  tubular  boilers  consuming 
bituminous  coal,  though  indeed  the  evil  might  be  remedied  by  block- 
ing some  of  the  tubes  up.  The  area  of  water-level  is  about  5  feet 
per  horse  power  in  land  boilers.  In  many  cases,  however,  it  is 
much  less ;  but  it  is  always  desirable  to  make  the  area  of  the  water- 
level  as  large  as  possible,  as,  when  it  is  contracted,  not  only  is  the 
water-level  subject  to  sudden  and  dangerous  fluctuations,  but  water 
is  almost  sure  to  be  carried  into  the  cylinder  with  the  steam,  in 
consequence  of  the  violent  agitation  of  the  water,  caused  by  the 
ascent  of  a  large  volume  of  steam  through  a  small  superficies.  It 
would  be  an  improvement  in  boilers,  we  think,  to  place  over  each 
furnace  an  inverted  vessel  immerged  in  the  water,  which  might 
catch  the  steam  in  its  ascent,  and  deliver  it  quietly  by  a  pipe  rising 
above  the  water-level.  The  water-level  would  thus  be  preserved 
from  any  inconvenient  agitation,  and  the  weight  of  water  within  the 
boiler  would  be  diminished  at  the  same  time  that  the  original  depth 
of  water  over  the  furnaces  was  preserved.  It  would  also  be  an 
improvement  to  make  the  sides  of  the  furnaces  of  marine  boilers 
sloping,  instead  of  vertical,  as  is  the  common  practice,  for  the  steam 
could  then  ascend  freely  at  the  instant  of  its  formation,  instead  of 
being  entangled  among  the  rivets  and  landings  of  the  plates,  and 
superinducing  an  overheating  of  the  plates  by  preventing  a  free 
access  of  the  water  to  the  metal. 

We  have,  in  the  following  table,  collected  a  few  of  the  principal 
results  of  experiments  made  on  steam  boilers. 


THE    STEAM    ENGINE. 

TABLE  I. 


215 


'5s 

:!! 

§ 

g 

1 

3 

NATURE  OF  THK  BOILERS  USED. 

it 

IS 
il 

,  at  Warwick. 

1 
I?!,. 

« 

I 

i 

of  llofM.  dePaml 

erinients. 

sh  boiler  at  the  East  L 
Water  Works,  1839. 

er  boiler  at  the  East 
Waterworks,  1839 

I3 

I 

|in 

n 

8  g 

1 

4^ 

Cylindrical 
with  inter- 
nal  flue. 

Wa^on. 

Wagon. 

Circular  or 
Hay  -stack. 

Locomo- 
tive. 

Cylindrical 
with  inter- 
nal flue. 

Wagon 
with  inter- 
nal flue. 

Total  area  of  heated  sur-  ) 
face  in  square  feet  ) 
Length  of  circuit  made  by  > 

962 
155 

152 
5066 

342-8 
72-5 

459 
52-8 

334-6 
7-0 

798 
83-1 

588 

78 

Area  of  fire  grates  in  square 

23-66 

23-33 

26-09 

35-10 

7-03 

14-25 

37-26 

Weight  of  fuel  burned  on  | 

each  square  foot  of  grate,  >- 

3-46 

4-00 

10-75 

20-34 

79-33 

46-82 

13-31 

Cub.  ft.  of  water  evaporated 
from  initial  temperature 

18-87 

16-44 

13-91 

14-11 

11-14 

Cubic  feet  of   water  eva- 

porated per   hour  from 

13-81 

13-79 

34-40 

90-7 

55-18 

initial  temperature  

Square  feet  of  heated  sur- 

face1 for  each  cubic  foot  of 
water     evaporated     per 

69-58 

11-00 

9-96 

5-06 

6-06 

17-17 

Square  feet  of  heated  sur- 

face for  each  square  foot 

40-65 

6-51 

13-13 

13-08 

47-59 

56-0 

15-78 

Pressure  of   steam  above 
the  atmosphere  in  Ibs.-  • 

42-2 

2-5 

3-68 

1-5 

50- 

15-45 

The  economical  effects  of  expansion  will  be  found  to  be  very 
clearly  exhibited  in  the  next  table.  The  duties  are  recorded  in  the 
fifth  line  from  the  top,  and  the  degree  of  expansion  in  the  bottom 
line.  It  will  be  observed,  that  the  order  in  which  the  different  en- 
gines stand  in  respect  of  superiority  of  duty  is  the  same  as  in  re- 
spect of  amount  of  expansion.  The  Holmbush  engine  has  a  duty 
of  140,484,848  Ibs.  raised  1  foot  by  1  cwt.  of  coals,  and  the  steam 
acts  expansively  over  '83  of  the  whole  stroke ;  while  the  water- 
works' Cornish  engine  has  only  a  duty  of  105,664,118  Ibs.,  and 
expands  the  steam  over  only  '687  of  the  whole  stroke.  Again, 
comparing  the  second  and  last  engines  together,  the  Albion  Mills 
engine  has  a  duty  of  25,756,752  Ibs.,  and  no  expansive  action. 
The  water-works'  engine,  again,  acts  expansively  over  one-half  of 
its  stroke,  and  has  an  increased  duty  of  46,602,333  Ibs.  Other 
causes,  of  course,  may  influence  these  comparisons,  especially  the 
last,  where  one  engine  is  a  double-acting  rotative  engine,  and  the 
other  a  single-acting  pumping  one  ;  but  there  can  be  no  doubt  that 
the  expansive  action  in  the  latter  is  the  principal  cause  of  its  more 
economical  performance. 

The  heating  surface  per  horse  power  allowed  by  some  engineers 
is  about  9  square  feet  in  wagon  boilers,  reckoning  the  total  sur- 
face as  effective  surface,  if  the  boilers  be  of  a  considerable  size ; 
but  in  the  case  of  small  boilers,  the  proportion  is  larger.  The  total 


216 


THE  PRACTICAL  MODEL  CALCULATOR. 


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i-H                »O       CO   CO                        I—           T-l 

injooCi  O      rH      CO  CO                    CO         GO         HO 
§*-£       GS}      gCO~                   r^H          A 

$~ 

Cornish  Engine, 
East  London 
Water  Works. 

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Holmbush,  Cornish, 
condensing  En- 
gine, single  acting 
for  pumping  water. 
Steam  acts  expan- 
sivel  yafter  the  first 
sixth  of  the  stroke. 
1836. 

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Engine,  Albion 
Mills,  London, 
date  1786. 
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Diameter  of  cylinder  in  inches  
Length  of  stroke  in  feet  
Number  of  strokes  per  minute  
Pressure  on  the  piston,  above  or  below 
in  Ibs.,  per  square  inch  
Weight  in  Ibs.  raised  one  foot  by  112 
Do.  do.  by  one  pound 
Effective  power  of  the  engine  at  time 
horse  power  
Efficiency  of  the  steam,  its  efficienc 
Mills  being  unity  
Efficiency  of  the  fuel,  its  efficiency  in 
being  unity  
Distance  of  the  piston  from  the  end  ol 
the  steam  is  cut  off  in  parts  of  the 

THE   STEAM   ENGINE.  217 

heating  surface  of  a  two  horse  power  wagon  boiler  is,  according 
to  Fitzgerald's  proportions,  30  square  feet,  or  15  ft.  per  horse 
power ;  whereas,  in  the  case  of  a  45  horse  power  boiler  the  total 
heating  surface  is  438  square  feet,  or  9-6  ft.  per  horse  power. 
The  capacity  of  steam  room  is  8}  cubic  feet  per  horse  power,  in 
the  two  horse  power  boiler,  and  5f  cubic  feet  in  the  20  horse  power 
boiler ;  and  in  the  larger  class  of  boilers,  such  as  those  suitable  for 
30  and  45  horse  power  engines,  the  capacity  of  the  steam  room 
does  not  fall  below  this  amount,  and  indeed  is  nearer  6  than  5f  cu- 
bic feet  per  horse  power.  The  content  of  water  is  18J  cubic  feet 
per  horse  power  in  the  two  horse  power  boiler,  and  15  cubic  feet 
per  horse  power  in  the  20  horse  power  boiler.  In  marine  boilers 
about  the  same  proportions  obtain  in  most  particulars.  The  ori- 
ginal boilers  of  one  or  two  large  steamers  were  proportioned 
with  about  half  a  square  foot  of  fire  grate  per  horse  power,  and  10 
square  feet  of  flue  and  furnace  surface,  reckoning  the  total  amount 
as  effective ;  but  in  the  boilers  of  other  vessels  a  somewhat  smaller 
proportion  of  heating  surface  was  adopted.  In  some  cases  we 
have  found  that,  in  their  marine  flue  boilers,  9  square  feet  of 
flue  and  furnace  surface  are  requisite  to  boil  off  a  cubic  foot  of 
water  per  hour,  which  is  the  proportion  that  obtains  in  some  land 
boilers  ;  but  inasmuch  as  in  modern  engines  the  nominal  considera- 
bly exceeds  the  actual  power,  they  allow  11  square  feet  of  heating 
surface  per  nominal  horse  power  in  their  marine  boilers,  and  they 
reckon,  as  effective  heating  surface,  the  tops  of  the  flues,  and  the 
whole  of  the  sides  of  the  flues,  but  not  the  bottoms.  They  have 
been  in  the  habit  of  allowing  for  the  capacity  of  the  steam  space 
in  marine  boilers  16  times  the  content  of  the  cylinder  ;  but  as  there 
are  two  cylinders,  this  is  equivalent  to  8  times  the  content  of  both 
cylinders,  which  is  the  proportion  commonly  followed  in  land  en- 
gines, and  which  agrees  very  nearly  with  the  proportion  of  between 
5  and  6  cubic  feet  of  steam  room  per  horse  power.  Taking,  for 
example,  an  engine  with  23  inches  diameter  of  cylinder  and  4  feet 
stroke,  which  will  be  18*4  horse  power — the  area  of  the  cylinder 
will  be  415-476  square  inches,  which,  multiplied  by  48,  the  number 
of  inches  in  the  stroke,  will  give  19942-848  for  the  capacity  of  the 
cylinder  in  cubic  inches ;  8  times  this  is  159542-784  cubic  inches, 
or  92-3  cubic  feet;  92-3  divided  by  18-4  is  rather  more  than  5  cu- 
bic feet  per  horse  power.  There  is  less  necessity,  however,  that 
the  steam  space  should  be  large  when  the  flow  of  steam  from  the 
boiler  is  very  uniform,  as  it  will  be  where  there  are  two  engines  at- 
tached to  the  boiler  at  right  angles  with  one  another,  or  where  the 
engines  work  at  a  great  speed,  as  in  the  case  of  locomotive  engines. 
A  high  steam  chest  too,  by  rendering  boiling  over  into  the  steam 
pipes,  or  priming  as  it  is  called,  more  difficult,  obviates  the  neces- 
sity for  so  large  a  steam  space ;  and  the  use  of  steam  of  a  high 
pressure,  worked  expansively,  has  the  same  operation  ;  so  that  in 
modern  marine  boilers,  of  the  tubular  construction,  where  the  whole 
of  these  modifying  circumstances  exist,  there  is  no  necessity  for  so 


218 


THE  PRACTICAL  MODEL  CALCULATOR. 


large  a  proportion  of  steam  room  as  5  or  6  cubic  feet  per  horse 
power,  and  about  half  that  amount  more  nearly  represents  the 
general  practice.  Many  allow  0-64  of  a  square  foot  per  nomi- 
nal horse  power  of  grate  bars  in  their  marine  boilers,  and  a  good 
effect  arises  from  this  proportion  ;  but  sometimes  so  large  an  area 
of  fire  grate  cannot  be  conveniently  got,  and  the  proportion  of 
half  a  square  foot  per  horse  power  seems  to  answer  very  well  in 
engines  working  with  some  expansion,  and  is  now  very  widely 
adopted.  With  this  allowance,  there  will  be  about  22  square  feet 
of  heating  surface  per  square  foot  of  fire  grate  ;  and  if  the  consump- 
tion of  fuel  be  taken  at  6  Ibs.  per  nominal  horse  power  per  hour, 
there  will  be  12  Ibs.  of  coal  consumed  per  hour  on  each  square  foot 
of  grate.  The  flues  of  all  flue  boilers  diminish  in  their  calorimeter 
as  they  approach  the  chimney ;  some  very  satisfactory  boilers  have 
been  made  by  allowing  a  proportion  of  0-6  of  a  square  foot  of  fire 
grate  per  nominal  horse  power,  and  making  the  sectional  area  of 
the  flue  at  the  largest  part  $&  of  the  area  of  fire  grate,  and '  the 
smallest  part,  where  it  enters  the  chimney,  ^Tth  of  the  area  of  the 
fire  grate ;  but  in  some  of  the  boilers  proportioned  on  this  plan  the 
maximum  sectional  area  is  only  Y^  or  -fa,  according  to  the  purposes 
of  the  boiler.  These  proportions  are  retained  whether  the  boiler  is 
flue  or  tubular,  and  from  14  to  16  square  feet  of  tube  surface  is  al- 
lowed per  nominal  horse  power ;  but  such  boilers,  although  they  may 
give  abundance  of  steam,  are  generally,  perhaps  needlessly,  bulky. 
We  shall  therefore  conclude  our  remarks  upon  the  subject  by 
introducing  a  table  of  the  comparative  evaporative  power  of  differ- 
ent kinds  of  coal,  which  will  prove  useful,  by  affording  data  for  the 
comparison  of  experiments  upon  different  boilers  when  different 
kinds  of  coal  are  used. 

TABLE  of  the  Comparative  Evaporative  Power  of  different  kinds 

of  Coal. 


No. 

Description  of  Coals. 

Water  evapo- 
rated per  Ib. 
of  Coals. 

1 

The  best  Welsh  

Us. 

9-493 

2- 

Anthracite  American     .      .            

9-14 

3 

The  best  small  Pittsburgh  

8-526 

4 

8-074 

5 

Pennsylvanian  ..             .         

10-45 

6 

Coke  from  Gas-works  

7-908 

7 
8 
9 
10 

Coke  and  Newcastle,  small,  £  and  £.... 
Welsh  and  Newcastle,  mixed  £  and  £... 
Derbyshire  and  small  Newcastle,  £  and  J 
Average  large  Newcastle       

7-897 
7-865 
7-710 
7-658 

11 

6-772 

12 

Blythe  Main   Northumberland    

6-600 

Strength  of  boilers. — The  extension  of  the  expansive  method  of 
employing  steam  to  boilers  of  every  denomination,  and  the  gradual 
introduction  in  connection  therewith  of  a  higher  pressure  than  for- 


THE    STEAM    ENGINE. 


219 


merly,  makes  the  question  of  the  strength  of  boilers  one  of  great 
and  increasing  importance.  This  topic  was  very  successfully  eluci- 
dated, a  few  years  ago,  by  a  committee  of  the  Franklin  Institute, 
Philadelphia,  and  we  shall  here  recapitulate  a  few  of  the  more  im- 
portant of  the  conclusions  at  which  they  arrived.  Iron  boiler  plate 
was  found  to  increase  in  tenacity  as  its  temperature  was  raised,  un- 
til it  reached  a  temperature  of  550°  above  the  freezing  point,  at 
which  point  its  tenacity  began  to  dimmish.  The  following  table 
exhibits  the  cohesive  strength  at  different  temperatures. 

At      32°  to  80°  the  tenacity  was  =  56,000  Ibs.,  or  l-7th  below  its  maximum. 


At  570° 
At  720° 
At  1050° 
At  1240° 
At  1317° 
At  3000°  iron  becomes  fluid. 


=  66,500  Ibs.,  the  maximum. 
=  55,000  Ibs.,  the  same  nearly  as  at  32°. 
=  32,000  Ibs.,  nearly  £  of  the  maximum. 
=  22,000  Ibs.,  nearly  J  of  the  maximum. 
=  9,000  Ibs.,  nearly  l-7th  of  the  maximum. 


The  difference  in  strength  between  strips  of  iron  cut  in  the  di- 
rection of  the  fibre,  and  strips  cut  across  the  grain,  was  found  to 
be  about  6  per  cent,  in  favour  of  the  former.  Repeated  piling  and 
welding  was  found  to  increase  the  tenacity  and  closeness  of  the 
iron,  but  welding  together  different  kinds  of  iron  was  found  to  give 
an  unfavourable  result ;  riveting  plates  was  found  to  occasion  a 
diminution  in  their  strength,  to  the  extent  of  about  one-third.  The 
accidental  overheating  of  a  boiler  was  found  to  reduce  its  strength 
from  65,000  Ibs.  to  45,000  Ibs.  per  square  inch.  Taking  into  ac- 
count all  these  contingencies,  it  appears  expedient  to  limit  the  ten- 
sile force  upon  boilers  in  actual  use  to  about  3000  Ibs.  per  square 
inch  of  iron. 

Copper  follows  a  different  law,  and  appears  to  diminish  in  strength 
by  every  addition  of  heat,  reckoning  from  the  freezing  point.  The 
square  of  the  diminution  of  strength  seems  to  keep  pace  with  the 
cube  of  the  temperature,  as  appears  by  the  following  table  : — 

TABLE  showing  the  Diminution  of  Strength  of  COPPER  Boiler 
Plates  by  additions  to  the  Temperature,  the  Cohesion  at  32°  being 
32,800  Ibs.  per  Square  Inch. 


No. 

Temperature 
above  32O. 

Diminution  of 
Strength. 

No. 

Temperature 
above  32O. 

Diminution  of 

Strength. 

1 

90° 

0-0175 

9 

660° 

0-3425 

2 

180 

0-0540 

10 

769 

0-4398 

3 

270 

0-0926 

11 

812 

0-4944 

4 

360 

0-1513 

1-2 

880 

0-5581 

5 

450 

0-2046 

13 

984 

0-6691 

6 

460 

0-2133 

14 

1000 

0-6741 

7 

513 

0-2446 

15 

1200 

0-8861 

8 

629 

0-2558 

16 

1300 

1-0000 

In  the  case  of  iron,  the  following  are  the  results  when  tabulated 
after  a  similar  fashion. 


220 


THE  PRACTICAL  MODEL  CALCULATOR. 


TABLE  of  Experiments  on  IRON  Boiler  Plate  at  High  Tempera- 
ture; the  Mean  Maximum  Tenacity  being  at  550°  =  65,000  Ibs. 
per  Square  Inch. 


Temperature 
observed. 

Diminution  of 
Tenacity  observed. 

Temperature 
observed. 

Diminution  of 
Tenacity  observed. 

550° 

0-0000 

824° 

0-2010 

570 

0-0869 

932 

0-3324 

696 

0-0899 

947 

0-3593 

600 

0-0964 

1030 

0-4478 

630 

0-1047 

1111 

0-5514 

562 

0-1155 

1155 

0-6000 

722 

0-1436 

1159 

0-6011 

732 

0-1491 

1187 

0-6352 

734 

0-1535 

1237 

0-6622 

766 

0-1589 

1245 

0-6715 

770 

0-1627 

1317 

0-7001 

The  application  of  stays  to  marine  boilers,  especially  in  those 
parts  of  the  water  spaces  which  lie  in  the  wake  of  the  furnace  bars, 
has  given  engineers  much  trouble ;  the  f  plate,  of  which  ordinary 
boilers  are  composed,  is  hardly  thick  enough  to  retain  a  stay  with 
security  by  merely  tapping  the  plate,  whereas,  if  the  stay  be  ri- 
veted, the  head  of  the  rivet  will  in  all  probability  be  soon  burnt 
away.  The  best  practice  appears  to  be  to  run  the  stays  used  for 
the  water  spaces  in  this  situation,  in  a  line  somewhat  beneath  the 
level  of  the  bars,  so  that  they  may  be  shielded  as  much  as  possible 
from  the  fire,  while  those  which  are  required  above  the  level  of  the 
bars  should  be  kept  as  nearly  as  possible  towards  the  crown  of  the 
furnace,  so  as  to  be  removed  from  the  immediate  contact  of  the  fire. 
Screw  bolts  with  a  fine  thread  tapped  into  the  plate,  and  with  a 
thin  head  upon  the  one  side,  and  a  thin  nut  made  of  a  piece  of 
boiler  plate  on  the  other,  appear  to  be  the  best  description  of  stay 
that  has  yet  been  contrived.  The  stays  between  the  sides  of  the 
boiler  shell,  or  the  bottom  of  the  boiler  and  the  top,  present  little 
difficulty  in  their  application,  and  the  chief  thing  that  is  to  be  at- 
tended to  is  to  take  care  that  there  be  plenty  of  them  ;  but  we  may 
here  remark  that  we  think  it  an  indispensable  thing,  when  there  is 
any  high  pressure  of  steam  to  be  employed,  that  the  furnace  crown 
be  stayed  to  the  top  of  the  boiler.  This,  it  will  be  observed,  is  done 
in  the  boilers  of  the  Tagus  and  Infernal ;  and  we  know  of  no  better 
specimen  of  staying  than  is  afforded  by  those  boilers. 


AREA   OF    STEAM    PASSAGES. 


RULE. — To  the  temperature  of  steam  in  the  boiler  add  the  con- 
stant increment  459 ;  multiply  the  sum  by  11025  ;  and  extract  the 
square  root  of  the  product.  Multiply  the  length  of  stroke  by  the 
number  of  strokes  per  minute ;  divide  the  product  by  the  square 
root  just  found ;  and  multiply  the  square  root  of  the  quotient  by 
the  diameter  of  the  cylinder ;  the  product  will  be  the  diameter  of 
the  steam 


THE    STEAM    ENGINE.  221 

Let  it  be  required  to  determine  the  diameter  of  the  steam  pas- 
sages in  an  engine  of  which  the  diameter  of  the  cylinder  is  48 
inches,  the  length  of  stroke  4J  feet,  and  the  number  of  strokes  per 
minute  26,  supposing  the  temperature  under  which  the  steam  is 
generated  to  be  250  degrees  of  Fahrenheit's  thermometer. 

Here  by  the  rule  we  get  ^11025(250  +  459)  =  2795-84 ;  the 
number  of  strokes  is  26,  and  the  length  of  stroke  4  J  feet ;  hence 

,— jjy— 

it  is  8  =  ^\2795-84  =  °'20456^  =  0-20456  x  48  =  9-819  inches  ; 
so  that  the  diameter  of  the  steam  passages  is  a  little  more  than  one- 
fifth  of  the  diameter  of  the  cylinder.  The  same  rule  will  answer 
for  high  and  low  pressure  engines,  and  also  for  the  passages  into 
the  condenser. 

LOSS  OF  FORCE  BY  THE  DECREASE  OF  TEMPERATURE  IN  THE  STEAM  PIPES. 

RULE. — From  the  temperature  of  the  surface  of  the  steam  pipes 
subtract  the  temperature  of  the  external  air ;  multiply  the  remain- 
der by  the  length  of  the  pipes  in  feet,  and  again  by  the  constant 
number  or  coefficient  1-68  ;  then  divide  the  product  by  the  diameter 
of  the  pipe  in  inches  drawn  into  the  velocity  of  the  steam  in  feet 
per  second,  and  the  quotient  will  express  the  diminution  of  tem- 
perature in  degrees  of  Fahrenheit's  thermometer. 

Let  the  length  of  the  steam  pipe  be  16  feet  and  its  diameter  5 
inches,  and  suppose  the  velocity  of  the  steam  to  be.  about  95  feet 
per  second,  what  will  be  the  diminution  of  temperature,  on  the  sup- 
position that  the  steam  is  at  250°  and  the  external  air  at  60°  of 
Fahrenheit  ? 

Here,  by  the  note  to  the  above  rule,  the  temperature  of  the  sur- 
face of  the  steam  pipe  is  250  —  250  x  0-05  =  237'5  ;  hence  we  get 
th  =  1-68x16(287-5 -60)  =  10-044  degrees. 

If  we  examine  the  manner  of  the  composition  of  the  above  equa- 
tion, it  will  be  perceived  that,  since  the  diameter  of  the  pipe  and 
the  velocity  of  motion  enter  as  divisors,  the  loss  of  heat  will  be  less 
as  these  factors  are  greater ;  but,  on  the  other  hand,  the  loss  of 
heat  will  be  greater  in  proportion  to  the  length  of  pipe  and  the 
temperature  of  the  steam.  Since  the  steam  is  reduced  from  a 
higher  to  a  lower  temperature  during  its  passage  through  the  steam 
pipes,  it  must  be  attended  with  a  corresponding  diminution  in  the 
elastic  force ;  it  therefore  becomes  necessary  to  ascertain  to  what 
extent  the  force  is  reduced,  in  consequence  of  the  loss  of  heat  that 
takes  place  in  passing  along  the  pipes.  This  is  an  inquiry  of  some 
importance  to  the  manufacturers  of  steam  engines,  as  it  serves  to 
guard  them  against  a  very  common  mistake  into  which  they  are 
liable  to  fall,  especially  in  reference  to  steamboat  engines,  where  it 
is  usual  to  cause  the  pipe  to  pass  round  the  cylinder,  instead  of 
carrying  it  in  the  shortest  direction  from  the  boiler,  in  order  to  de- 
crease the  quantity  of  surface  exposed  to  the  cooling  effect  of  the 
atmosphere. 

T2 


222  THE   PRACTICAL   MODEL   CALCULATOR. 

RULE. — From  the  temperature  of  the  surface  of  the  steam  pipe 
subtract  the  temperature  of  the  external  air ;  multiply  the  remain- 
der by  the  length  of  the  pipe  in  feet,  and  again  by  the  constant 
fractional  coefficient  0-00168 ;  divide  the  product  by  the  diameter 
of  the  pipe  in  inches  drawn  into  the  velocity  of  steam  in  feet  per 
second,  and  subtract  the  quotient  from  unity ;  then  multiply  the 
difference  thus  obtained  by  the  elastic  force  corresponding  to  the 
temperature  of  steam  in  the  boiler,  and  the  product  will  be  the 
elastic  force  of  the  steam  as  reduced  by  cooling  in  passing  through 
the  pipes. 

Let  the  dimensions  of  the  pipe,  the  temperature  of  the  steam, 
and  its  velocity  through  the  passages,  be  the  same  as  in  the  pre- 
ceding example,  what  will  be  the  quantity  of  reduction  in  the  elastic 
force  occasioned  by  the  effect  of  cooling  in  traversing  the  steam 
pipe  ? 

Since  the  elastic  force  of  the  steam  in  the  boiler  enters  the  equa- 
tion from  which  the  above  rule  is  deduced,  it  becomes  necessary  in 
the  first  place  to  calculate  its  value ;  and  this  is  to  be  done  by  a 
rule  already  given,  which  answers  to  the  case  in  which  the  tempera- 
ture is  greater  than  212° ;  thus  we  have 

250  x  1-69856  =  424-640 
Constant  number  =  205-526  add 

•  Sum  =  630-166 log.  2-79945     . 

Constant  divisot  =  333 log.  2^522444  subtract 

0-277011  x  6-42  =  1-778410, 
which  is  the  logarithm  of  60-036  inches  of  mercury. 

Again,  we  have  250  —  0-05  x  250  =  237'5;  consequently,  by 
multiplying  as  directed  in  the  rule,  we  get  237*5  X  0-00168  X  16 
=  6-384,  which  being  divided  by  95  X  5  =  475,  gives  0-01344 ;  and 
by  taking  this  from  unity  and  multiplying  the  remainder  by  the 
elastic  force  as  calculated  above,  the  value  of  the  reduced  elastic 
force  becomes 

/  =  60-036  (1  -  0-01344)  =  59-229  inches  of  mercury. 

The  loss  of  force  is  therefore  60-036  —  59-229  =  0-807  inches  of 
mercury,  which  amounts  to  ^th  part  of  the  entire  elastic  force  of 
the  steam  in  the  boiler  as  generated  under  the  given  temperature, 
being  a  quantity  of  sufficient  importance  to  claim  the  attention  of 
our  engineers. 

FEED    WATER. 

The  quantity  of  water  required  to  supply  the  waste  occasioned 
by  evaporation  from  a  boiler,  or,  as  it  is  technically  termed,  the 
"  feed  water"  required  by  a  boiler  working  with  any  given  pressure, 
is  easily  determinable.  For,  since  the  relative  volumes  of  water 
and  steam  at  any  given  pressure  are  known,  it  becomes  necessary 
merely  to  restore  the  quantity  of  water  by  the  feed  pump  equiva- 


THE    STEAM    ENGINE.  223 

lent  to  that  abstracted  in  the  form  of  steam,  which  the  known  rela- 
tion of  the  density  to  the  pressure  of  the  steam  renders  of  easy 
accomplishment.  In  practice,  however,  it  is  necessary  that  the 
feed  pump  should  be  able  to  supply  a  much  larger  quantity  of  water 
than  what  theory  prescribes,  as  a  great  waste  of  water  sometimes 
occurs  from  leakage  or  priming,  and  it  is  necessary  to  provide 
against  such  contingencies.  The  feed  pump  is  usually  made  of 
such  dimensions  as  to  be  capable  of  supplying  3J  times  the  water 
that  the  boiler  will  evaporate,  and  in  low  pressure  engines,  where 
the  cylinder  is  double  acting  and  the  feed  pump  single  acting,  this 
proportion  will  be  maintained  by  making  the  pump  a  240th  of  the 
capacity  of  the  cylinder.  In  low  pressure  engines  the  pressure  in 
the  boiler  may  be  taken  at  5  Ibs.  above  the  pressure  of  the  atmo- 
sphere, or  20  Ibs.  in  all ;  and  as  high  pressure  steam  is  merely  low 
pressure  steam  compressed  into  a  smaller  compass,  the  size  of  the 
feed  pump  relatively  to  the  size  of  the  cylinder  must  obviously  vary 
in  the  direct  proportion  of  the  pressure.  If,  then,  the  feed  pump 
be  l--240th  of  the  capacity  of  the  cylinder  when  the  total  pressure 
of  the  steam  is  20  Ibs.,  it  must  be  l-120th  of  the  capacity  of  the 
cylinder  when  the  total  pressure  of  the  steam  is  40  Ibs.,  or  25  Ibs. 
above  the  atmosphere.  This  law  of  variation  is  expressed  by  the  fol- 
lowing rule,  which  gives  the  capacity  of  feed  pump  proper  for  all 
pressures  : — Multiply  the  capacity  of  the  cylinder  in  cubic  inches  by 
the  total  pressure  of  the  steam  in  Ibs.  per  square  inch,  or  the  pressure 
in  Us.  per  square  inch  on  the  safety  valve,  plus  15,  and  divide  the 
product  by  4800;  the  quotient  is  the  capacity  of  the  feed  pump  in 
cubic  inches,  when  the  feed  pump  is  single  acting  and  the  engine 
double  acting.  If  the  feed  pump  be  double  acting,  or  the  engine 
single  acting,  the  capacity  of  the  pump  must  be  just  one-half  what 
is  given  by  this  rule. 

CONDENSING   WATER. 

It  was  found  that  the  most  beneficial  temperature  of  the  hot 
well  was  100  degrees.  If,  therefore,  the  temperature  of  the 
steam  be  212°,  and  the  latent  heat  1000°,  then  1212°  may  be 
taken  to  represent  the  heat  contained  in  the-  steam,  or  1112° 
if  we  deduct  the  temperature  of  the  hot  well.  If  the  tempera- 
ture of  the  injection  water  be  50°,  then  50  degrees  of  cold  are 
available  for  the  abstraction  of  heat,  and  as  the  total  quantity  of 
heat  to  be  abstracted  is  that  requisite  to  raise  the  quantity  of  water 
in  the  steam  1112  degrees,  or  1112  times  that  quantity,  one  degree, 
it  would  raise  one-fiftieth  of  this,  or  22-24  times  the  quantity  of 
water  in  the  steam,  50  degrees.  A  cubic  inch  of  water,  therefore, 
raised  into  steam,  will  require  22-24  cubic  inches  of  water  at  50 
degrees  for  its  condensation,  and  will  form  therewith  23-24  cubic 
inches  of  hot  water  at  100  degrees.  It  has  been  a  practice  to 
allow  about  a  wine  pint  (28-9  cubic  inches)  of  injection  water  for 
every  cubic  inch  of  water  evaporated  from  the  boiler.  The  usual 
capacity  for  the  cold  water  pump  is  ^th  of  the  capacity  of  the 
cylinder,  which  allows  some  water  to  run  to  waste.  As  a  maximum 


224  THE   PRACTICAL    MODEL   CALCULATOR. 

effect  is  obtained  when  the  temperature  of  the  hot  well  is  about 
100°,  it  will  not  be  advisable  to  reduce  it  below  that  temperature 
in  practice.  With  the  superior  vacuum  due  to  a  temperature  of  70° 
or  80°  the  admission  of  so  much  cold  water  into  the  condenser 
becomes  necessary, — and  which  has  afterwards  to  be  pumped  out 
in  opposition  to  the  pressure  of  the  atmosphere, — so  that  the  gain  in 
the  vacuum  does  not  equal  the  loss  of  power  occasioned  by  the 
additional  load  upon  the  pump,  and  there  is,  therefore,  a  clear  loss 
by  the  reduction  of  the  temperature  below  100°,  if  such  reduction 
be  caused  by  the  admission  of  an  additional  quantity  of  water.  If 
the  reduction  of  temperature,  however,  be  caused  by  the  use  of 
colder  water,  there  is  a  gain  produced  by  it,  though  the  gain  will 
within  certain  limits  be  greater,  if  advantage  be  taken  of  the  low- 
ness  of  the  temperature  to  dimmish  the  quantity  of  injection. 


SAFETY  VALVES. 

RULE.  —  Add  459  to  the  temperature  of  the  steam  in  degrees 
of  Fahrenheit  ;  divide  the  sum  by  the  product  of  the  elastic  force 
of  the  steam  in  inches  of  mercury,  into  its  excess  above  the  weight 
of  the  atmosphere  in  inches  of  mercury  ;  multiply  the  square  root 
of  the  quotient  by  -0653  ;  multiply  this  product  by  the  number  of 
cubic  feet  per  hour  of  water  evaporated,  and  this  last  product  is 
the  theoretical  area  of  the  orifice  of  the  safety  valve  in  square 
inches. 

To  apply  this  to  an  example  —  which,  however,  it  must  be  remem- 
bered, will  give  a  result  much  too  small  for  practice. 

Required  the  least  area  of  a  safety  valve  of  a  boiler  suited  for  a 
250  horse  power  engine,  working  with  steam  6  Ibs.  more  than  the 
atmosphere  on  the  square  inch. 

In  this  case  the  total  pressure  is  equal  to  21  Ibs.  per  square 
inch  ;  and  as  in  round  numbers  one  pound  of  pressure  is  equal  to 
about  two  inches  of  mercury,  it  follows  that  /  =  42  inches  of 
mercury. 

It  will  be  necessary  to  calculate  t  from  formula  (S)  already  given. 
The  operation  is  as  follows  :  — 

log.  42  -4-  6-42  =  1-623249  •+  6-42  =  0-252842 
constant  co-efficient  =  196  2-292363 

$^545205 

natural  number  =  350-92 
constant  temperature  =  121 

t  =  2~29T92~ 

229-92 


/459  +t  / 

therefore  4=W  =  4 


(f)  42  X  12 

=  1-168; 
therefore  x  =  -0653  x  1-168  x  N  =  -0757  N. 


THE    6Tr  AM    ENGINE.  225 

We  have  stated  in  a  former  part  of  this  v»ork  that  a  cubic  foot 
of  water  evaporated  per  hour  is  equivalent  to  one  horse  power; 
therefore  in  this  case  N  =  250  and  x  —  18-925  sq.  in. 

As  another  example.  Required  the  proper  area  of  the  safety 
valve  of  a  boiler  suited  to  an  engine  of  500  horse  power,  when  it 
is  wished  that  the  steam  should  never  acquire  an  elastic  force 
greater  than  60  Ibs.  on  the  square  inch  above  the  atmosphere. 

In  this  case  the  whole  elastic  force  of  the  steam  is  75Hbs.  ;  and 
as  1  pound  corresponds  in  round  numbers  to  2  inches  of  mercury, 
it  follows  that  /  =  150.  It  will  be  necessary  to  calculate  the 
temperature  corresponding  to  this  force.  The  operation  is  as 
follows  :  — 

Log.  150  +•  6-42  =  2-176091  -r-  642  =    -338955 
constant  co-efficient  =     196  log.  2-292363  add 

natural  number  =  427-876  2-631318 

constant  temperature  =  121 

required  temperature        306*876  degrees  of  Fahrenheit's  scale 

459  +  t     _  459  +  306-876  _     765-876    _  765-896 
therefore  -  _  30)  -  150  x  120  -:    18000 


=  -043549  ;  therefore  ^  V  *042549  =  *20628- 


Hence  the  required  area  =  -0653  X  -20628  X  500  =  -01347  X 
500  =  6-735  square  inches. 

If  the  area  of  the  safety  valve  of  a  boiler  suited  for  an  engine 
of  500  horse  power  be  required,  when  it  is  wished  the  steam  should 
never  acquire  a  greater  temperature  than  300°,  it  will  be  necessary 
to  calculate  the  elastic  force  corresponding  to  this  temperature  ;  and 
by  formula  for  this  purpose,  the  required  area  =  -0653  X  -231  X 
500  =  -0151  X  500  =  7*55  square  inches.  It  will  be  perceived 
from  these  examples  that  the  greater  the  elasticity  and  the  higher 
the  corresponding  temperature  the  less  is  the  area  of  the  safety 
valve.  This  is  just  as  might  have  been  expected,  for  then  the 
steam  can  escape  with  increased  velocity.  We  may  repeat  that  the 
results  we  have  arrived  at  are  much  less  than  those  used  in  practice. 
For  the  sake  of  safety,  the  orifices  of  the  safety  valve  are  inten- 
tionally made  much  larger  than  what  theory  requires  ;  usually  ^  of 
a  square  inch  per  horse  power  is  the  ordinary  proportion  allowed 
in  the  case  of  low  pressure  engines. 

THE    SLIDE   VALVE. 

The  four  following  practical  rules  are  applicable  alike  to  short 
slide  and  long  D  valves. 

RULE  I.  —  To  find  how  much  cover  must  be  given  on  the  steam 
side  in  order  to  cut  the  steam  off  at  any  given  part  of  the  stroke.  — 
From  the  length  of  the  stroke  of  the  piston,  subtract  the  length  of 
that  part  of  the  stroke  that  is  to  be  made  before  the  steam  is  cut 
off.  Divide  the  remainder  by  the  length  of  the  stroke  of  the 

15 


OF   THK 

TTNTTVERSITY 


226  THE   PRACTICAL    MODEL    CALCULATOR. 

piston,  and  extract  the  square  root  of  the  quotient.  Multiply  the 
square  root  thus  found  by  half  the  length  of  the  stroke  of  the  valve, 
and  from  the  product  take  half  the  lead,  and  the  remainder  will  be 
the  cover  required. 

RULE  II. —  To  find  at  what  part  of  the  stroke  any  given  amount 
of  cover  on  the  steam  side  will  cut  off  the  steam. — Add  the  cover 
on  the  steam  side  to  the  lead ;  divide  the  sum  by  half  the  length 
>f  stroke  of  the  valve.  In  a  table  of  natural  sines  find  the  arc 
tfhose  sine  is  equal  to  the  quotient  thus  obtained.  To  this  arc  add 
90°,  and  from  the  sum  of  these  two  arcs  subtract  the  arc  whose 
cosine  is  equal  to  the  cover  on  the  steam  side  divided  by  half  the 
stroke  of  the  valve.  Find  the  cosine  of  the  remaining  arc,  add  1 
to  it,  and  multiply  the  sum  by  half  the  stroke  of  the  piston,  and 
the  product  is  the  length  of  that  part  of  the  stroke  that  will  be 
made  by  the  piston  before  the  steam  is  cut  off. 

RULE  III. —  To  find  how  much  before  the  end  of  the  stroke,  the 
exhaustion  of  the  steam  in  front  of  the  piston  will  be  cut  off. — To 
the  cover  on  the  steam  side  add  the  lead,  and  divide  the  sum  by 
half  the  length  of  the  stroke  of  the  valve.  Find  the  arc  whose  sine 
is  equal  to  the  quotient,  and  add  90°  to  it.  Divide  the  cover  on 
the  exhausting  side  by  half  the  stroke  of  the  valve,  and  find  the  arc 
whose  cosine  is  equal  to  the  quotient.  Subtract  this  arc  from  the 
one  last  obtained,  and  find  the  cosine  of  the  remainder.  Subtract 
tfcis  cosine  from  2,  and  multiply  the  remainder  by  half  the  stroke 
of  the  piston.  The  product  is  the  distance  of  the  piston  from  the 
end  of  its  stroke  when  the  exhaustion  is  cut  off. 

RULE  IV. —  To  find  how  far  the  piston  is  from  the  end  of  its 
stroke,  when  the  steam  that  is  propelling  it  by  expansion  is  allowed 
to  escape  to  the  condenser. — To  the  cover  on  the  steam  side  add  the 
lead,  divide  the  sum  by  half  the  stroke  of  the  valve,  and  find  the 
arc  whose  sine  is  equal  to  the  quotient.  Find  the  arc  whose  cosine 
is  equal  to  the  cover  on  the  exhausting  side,  divided  by  half  the 
stroke  of  the  valve.  Add  these  two  arcs  together,  and  subtract 
90°.  Find  the  cosine  of  th£  residue,  subtract  it  from  1,  and  mul- 
tiply the  remainder  by  half  the  stroke  of  the  piston.  The  product 
is  the  distance  of  the  piston  from  the  end  of  its  stroke,  when  the 
steam  that  is  propelling  it  is  allowed  to  escape  to  the  condenser. 
In  using  these  rules,  all  the  dimensions  are  to  be  taken  in  inches, 
and  the  answers  will  be  found  in  inches  also. 

From  an  examination  of  the  formulas  we  have  given  on  this 
subject,  it  will  be  perceived  (supposing  that  there  is  no  lead)  that 
the  part  of  the  stroke  where  the  steam  is  cut  off,  is  determined  by 
the  proportion  which  the  cover  on  the  steam  side  bears  to  the 
length  of  the  stroke  of  the  valve :  so  that  in  all  cases  where  the 
cover  bears  the  same  proportion  to  the  length  of  the  stroke  of  the 
valve,  the  steam  will  be  cut  off  at  the  same  part  of  the  stroke  of 
the  piston. 

In  the  first  line,  accordingly,  of  Table  L,  will  be  found  eight 
iift'erent  parts  of  the  stroke  of  the  piston  designated ;  and  directly 


THE    STEAM   ENGINE. 


227 


below  each,  in  the  second  line,  is  given  the  quantity  of  cover  requi- 
site to  cause  the  steam  to  be  cut  off  at  that  particular  part  of  the 
stroke.  The  different  sizes  of  the  cover  are  given  in  the  second 
line,  in  decimal  parts  of  the  length  of  the  stroke  of  the  valve ;  so 
that,  to  get  the  quantity  of  cover  corresponding  to  any  of  the  given 
degrees  of  expansion,  it  is  only  necessary  to  take  the  decimal  in 
the  second  line,  which  stands  under  the  fraction  in  the  first,  that 
marks  the  degree  of  expansion,  and  multiply  that  decimal  by  the 
length  you  intend  to  make  the  stroke  of  the  valve.  Thus,  suppose 
you  have  an  engine  in  which  you  wish  to  have  the  steam  cut  off 
when  the  piston  is  a  quarter  of  the  length  of  its  stroke  from  the 
end  of  it,  look  in  the  table,  and  you  will  find  in  the  third  column 
from  the  left,  J.  Directly  under  that,  in  the  second  line,  you  have 
the  decimal  -250.  Suppose  that  you  think  18  inches  will  be  a  con- 
venient length  for  the  stroke  of  the  valve,  multiply  the  decimal 
•250  by  18,  which  gives  4J.  Hence  we  learn  that  with  an  18  inch 
stroke  for  the  valve,  4  J  inches  of  cover  on  the  steam  side  will  cause 
the  steam  to  be  cut  off  when  the  piston  has  still  a  quarter  of  its 
stroke  to  perform. 

Half  the  stroke  of  the  valve  must  always  be  at  least  equal  to  the 
cover  on  the  steam  side  added  to  the  breadth  of  the  port.  By  the 
"breadth"  of  the  port,  we  mean  its  dimension  in  the  direction  of 
the  valve's  motion ;  in  short,  its  perpendicular  depth  when  the 
cylinder  is  upright.  The  words  "cover"  and  "lap"  are  synony- 
Consequently,  as  the  cover,  in  this  case,  must  be  4J  inches, 


mous. 


and  as  half  the  stroke  of  the  valve  is  9  inches,  the  breadth  of  the 
port  cannot  be  more  than  (9  —  4J  =  4J)  4J  inches.  If  this 
breadth  of  port  is  not  enough,  we  must  increase  the  stroke  of  the 
valve  ;  by  which  means  we  shall  get  both  the  cover  and  the  breadth 
of  the  port  proportionally  increased.  Thus,  if  we  make  the  length 
of  valve  stroke  20  inches,  we  shall  have  for  the  cover  *250  x  20  =  5 
inches,  and  for  the  breadth  of  the  port  10  —  5  =  5  inches. 

TABLE  I. 


Distance  of  the  piston  from  "1 

8 

A 

4 

A 

2 

the   termination   of   its 

2? 

2? 

stroke,  when  the  steam  [• 

or 

^ 

or 

5 

or 

or 

or 

A 

is  cut  off,  in  parts  of  the  j 

length  of  its  stroke.        j 

i 

4 

i 

i 

A 

Cover  on  the  steam  side  of  "] 

the    valve,    in    decimal  1 
parts  of  the  length  of  its  j 

•289 

•270 

•250 

•228 

•204 

•177 

•144 

•102 

stroke. 

This  table,  as  we  have  already  intimated,  is  computed  on  the 
supposition  that  the  valve  is  to  have  no  lead  ;  but,  if  it  is  to  have 
lead,  all  that  is  necessary  is  to  subtract  half  the  proposed  lead  from 
the  cover  found  from  the  table,  and  the  remainder  will  be  the 


228 


THE   PRACTICAL   MODEL   CALCULATOR. 


proper  quantity  of  cover  to  give  to  the  valve.  Suppose  that,  in 
the  last  example,  the  valve  was  to  have  J  inch  of  lead,  we  would 
subtract  |  inch  from  the  5  inches  found  for  the  cover  by  the  table : 
that  would  leave  4|  inches  for  the  quantity  of  cover  that  the  valve 
ought  to  have. 

TABLE  II. 


Length  of 
the  stroke 

Cover  required  on  the  steam  side  of  the  valve  to  cut  the  steam  off  at  any  of  the 
under-noted  parts  of  the  stroke. 

Inches. 

1 

£ 

i 

ft 

1 

1 

& 

.  A 

24 

6-94 

6-48 

6-00 

5-47 

4-90 

4-25 

3-47 

2-45 

23A 

6-79 

6-34 

5-88 

5-36 

4-79 

4-16 

3-39 

2-39 

23 

6-65 

6-21 

5-75 

5-24 

4-69 

4-07 

3-32 

2-34 

22i 

6-50 

6-07 

5-62 

5-13 

4-59 

3-98 

3-25 

2-29 

22 

6-36 

5-94 

5-50 

5-02 

4-49 

3-89 

3-13 

2-24 

21| 

6-21 

5-80 

5-38 

4-90 

4-39 

3-80 

3-10 

2-19 

21 

6-07 

5-67 

5-25 

4-79 

4-28 

3-72 

3-03 

2-14 

20i 

5-92 

5-53 

5-12 

4-67 

4-18 

3-63 

2-96 

2-09 

20* 

5-78 

5-40 

5-00 

4-56 

4-08 

3-54 

2-89 

2-04 

•       19£ 

5-64 

5-26 

4-87 

4-45 

3-98 

3-45 

2-82 

1-99 

<       19 

5-49 

5-13 

4-75 

4-33 

3-88 

3-36 

2-74 

1-94 

18£ 

5-34 

4-99 

4-62 

4-22 

3-77 

3-27 

2-67 

1-88 

18 

5-20 

4-86 

4-50 

4-10 

3-67 

3-19 

2-60 

1-83 

!7* 

6-06 

4-72 

4-37 

3-99 

3-57 

3-10 

2-53 

1-78 

17 

4-91 

4-59 

4-25 

3-88 

3-47 

3-01 

2-45 

1-73 

16£ 

4-77 

4-45 

4-12 

3-76 

3-36 

2-92 

2-38 

1-68 

16 

4-62 

4-32 

4-00 

3-65 

3-26 

2-83 

2-31 

1-63 

15| 

448 

4-18 

3-87 

3-53 

3-16 

2-74 

2-24 

1-58 

15 

4-33 

4-05 

3-75 

3-42 

3-06 

2-65 

2-16 

1-53 

14£ 

4-19 

3-91 

3-62 

3-31 

2-96 

2-57 

2-09 

148 

14 

4-05 

3-78 

3-50 

3-19 

2-86 

2-48 

2-02 

143 

1*J 

3-90 

3-64 

3-37 

3-08 

2-75 

2-39 

1-95 

1-37 

13 

3-76 

3-51 

3-25 

2-96 

2-65 

2-30 

1-88 

1-32 

12* 

3-61 

3-37 

3-12 

2-85 

2-55 

2-21 

1-80 

1-27 

12 

3-47 

3-24 

3-00 

2-74 

2-45 

2-12 

1-73 

1-22 

11* 

3-32 

3-10 

2-87 

2-62 

2-35 

2-03 

1-66 

1-17 

11 

3-18 

2-97 

2-75 

2-51 

2-24 

1-95 

1-58 

1-12 

10£ 

3-03 

2-83 

2-62 

2-39 

2-14 

1-86 

1-51 

1-07 

10 

2-89 

2-70 

2-50 

2-28 

2-04 

1-77 

1-44 

1-02 

9* 

2-65 

2-56 

2-37 

2-17 

1-93 

1-68 

1-32 

•96 

9 

2-60 

2-43 

2-25 

2-05 

1-84 

1-59 

1-30 

•92 

8* 

2-46 

2-29 

2-12 

1-94 

1-73 

1-50 

1-23 

•86 

8 

2-31 

2-16 

2-00 

1-82 

1-63 

1-42 

1-15 

•81 

7* 

2-16 

2-02 

1-87 

1-71 

1-53 

1-33 

1-08 

•76 

7 

2-02 

1-89 

•75 

1-60 

1-43 

1-24 

1-01 

•71 

6* 

1-88 

1-75 

•62 

1-48 

1-32 

1-15 

•94 

•66 

6 

1-73 

1-62 

•50 

1-37 

1-22 

1-06 

•86 

•61 

5* 

1-58 

1-48 

•37 

1-25 

1-12 

•97 

•79 

•56 

5 

1-44 

1-35 

•25 

1-14 

1-02 

•88 

•72 

•51 

4* 

1-30 

1-21 

•12 

1-03 

•92 

•80 

•65 

•46 

4 

1-16 

1-08 

1-00 

•91 

•82 

•71 

•58 

41 

3* 

1-01 

•94 

•87 

•80 

•71 

•62 

•50 

•35 

3 

•86 

•81 

•75 

•68 

•61 

•53 

•44 

•30 

111 


Table  II.  is  an  extension  of  Table  I.  for  the  purpose  of  obviating, 
most  cases,  the  necessity  of  even  the  very  small  degree  of 
trouble  required  in  multiplying  the  stroke  of  the  valve  by  one  of • 
me  decimals  in  Table  I.  The  first  line  of  Table  II.  consists,  as  in 
Table  L,  of  eight  fractions,  indicating  the  various  parts  of  the  stroke 


THE   STEAM   ENGINE.  229 

at  which  the  steam  may  be  cut  off.  The  first  column  on  the  left 
hand  consists  of  various  numbers  that  represent  the  different 
lengths  that  may  be  given  to  the  stroke  of  the  valve,  diminishing, 
by  half-inches,  from  24  inches  to  3  inches.  Suppose  that  you  wish 
the  steam  cut  off  at  any  of  the  eight  parts  of  the  stroke  indicated 
in  the  first  line  of  the  table,  (say  at  ^  from  the  end  of  the  stroke,) 
you  find  J  at  the  top  of  the  sixth  column  from  the  left.  Look  for 
the  proposed  length  of  stroke  of  the  valve  (say  17  inches)  in  the 
first  column  on  the  left.  From  IT,  in  that  column,  run  along  the 
line  towards  the  right,  and  in  the  sixth  column,  and  directly  under 
the  J  at  the  top,  you  will  find  3-47,  which  is  the  cover  required  to 
cause  the  steam  to  be  cut  off  at  J  from  the  end  of  the  stroke,  if  the 
valve  has  no  lead.  If  you  wish  to  give  it  lead,  (say  J  inch,)  sub- 
tract the  half  of  that,  or  J  =  '125  inch  from  3-47,  and  you  will  have 
3-47  —  -125  =  3-345  inches,  the  quantity  of  cover  that  the  valve 
should  have. 

To  find  the  greatest  breadth  that  we  can  give  to  the  port  in  this 
case,  we  have,  as  before,  half  the  length  of  stroke,  8J— 3-345=5-155 
inches,  which  is  the  greatest  breadth  we  can  give  to  the  port  with 
this  length  of  stroke.  -  It  is  scarcely  necessary  to  observe  that  it  is 
not  at  all  essential  that  the  port  should  be  so  broad  as  this  ;  indeed, 
where  great  length  of  stroke  in  the  valve  is  not  inconvenient,  it  is 
always  an  advantage  to  make  it  travel  farther  than  is  just  neces- 
sary to  make  the  port  full  open;  because,  when  it  travels  farther, 
both  the  exhausting  and  steam  ports  are  more  quickly  opened,  so 
as  to  allow  greater  freedom  of  motion  to  the  steam. 

The  manner  of  using  this  table  is  so  simple,  that  we  need  not 
trouble  the  reader  with  more  examples.  We  pass  on,  therefore,  to 
explain  the  use  of  Table  III. 

Suppose  that  the  piston  of  a  steam  engine  is  making  its  down- 
ward stroke,  that  the  steam  is  entering  the  upper  part  of  the  cylin- 
der by  the  upper  steam-port,  and  escaping  from  below  the  piston 
by  the  lower  exhausting-port;  then,  if  (as  is  generally  the  case) 
the  slide  valve  has  some  cover  on  the  steam  side,  the  upper  port 
will  be  closed  before  the  piston  gets  to  the  bottom  of  the  stroke, 
and  the  steam  above  then  acts  expansively,  while  the  communica- 
tion between  the  bottom  of  the  cylinder  and  the  condenser  still 
continues  open,  to  allow  any  vapour  from  the  condensed  water  in 
the  cylinder,  or  any  leakage  past  the  piston,  to  escape  into  the 
condenser ;  but,  before  the  piston  gets  to  the  bottom  of  the  cylin- 
der, this  passage  to  the  condenser  will  also  be  cut  off  by  the  valve 
closing  the  lower  port.  Soon  after  the  lower  port  is  thus  closed, 
the  upper  port  will  be  opened  towards  the  condenser,  so  as  to  allow 
the  steam  that  has  been  acting  expansively  to  escape'.  Thus,  be- 
fore the  piston  has  completed  its  stroke,  the  propelling  power  is 
removed  from  behind  it,  and  a  resisting  power  is  opposed  before  it, 
arising  from  the  vapour  in  the  cylinder,  which  has  no  longer  any 
passage  open  to  the  condenser.  It  is  evident,  that  if  there  is  no 
cover  on  the  exhausting  side  of  the  valve,  the  exhausting  port  before 


230  THE   PRACTICAL   MODEL    CALCULATOR. 

the  piston  will  be  closed,  and  the  one  behind  it  opened,  at  the  same 
time ;  but,  if  there  is  any  cover  on  the  exhausting  side,  the  port 
before  the  piston  will  be  closed  before  that  behind  it  is  opened;  and 
the  interval  between  the  closing  of  the  one  and  the  opening  of 
the  other  will  depend  on  the  quantity  of  cover  on  the  exhausting 
side  of  the  valve.  Again,  the  position  of  the  piston  in  the  cylin- 
der, when  these  ports  are  closed  and  opened  respectively,  will 
depend  on  the  quantity  of  cover  that  the  valve  has  on  the  steam 
side.  If  the  cover  is  large  enough  to  cut  the  steam  off  when  the 
piston  is  yet  a  considerable  distance  from  the  end  of  its  stroke, 
these  ports  will  be  closed  and  opened  at  a  proportionably  early  part 
of  the  stroke ;  and  when  it  is  attempted  to  obtain  great  expansion 
by  the  slide-valve  alone,  without  an  expansion-valve,  considerable 
loss  of  power  is  incurred  from  this  cause. 

Table  III.  is  intended  to  show  the  parts  of  the  stroke  where,  un- 
der any  given  arrangement  of  slide  valve,  these  ports  close  and  open 
respectively,  so  that  thereby  the  engineer  may  be  able  to  estimate 
how  much  of  the  efficiency  of  the  engine  he  loses,  while  he  is  trying 
to  add  to  the  power  of  the  steam  by  increasing  the  expansion  in 
this  manner.  In  the  table,  there  are  eight  double  columns,  arid  at 
the  heads  of  these  columns  are  eight  fractions,  as  before,  represent- 
ing so  many  different  parts  of  the  stroke  at  which  the  steam  may 
be  supposed  to  be  cut  off. 

In  the  left-hand  single  column  in  each  double  one,  are  four  deci- 
mals, which  represent  the  distance  of  the  piston  (in  terms  of  the 
length  of  its  stroke)  from  the  end  of  its  stroke  when  the  exhausting- 
port  before  it  is  opened,  corresponding  with  the  degree  of  expansion 
indicated  by  the  fraction  at  the  top  of  the  double  column  and  the 
cover  on  the  exhausting  side  opposite  to  these  decimals  respectively 
in  the  left-hand  column.  The  right-hand  single  column  in  each 
double  one  contains  also  each  four  decimals,  which  show  in  the  same 
way  at  what  part  of  the  stroke  the  exhausting-port  behind  the  pis- 
ton is  opened.  A  few  examples  will,  perhaps,  explain  this  best. 

Suppose  we  have  an  engine  in  which  the  slide  valve  is  made  to 
cut  the  steam  off  when  the  piston  is  l-3d  from  the  end  of  its  stroke, 
and  that  the  cover  on  the  exhausting  side  of  the  valve  is  l-8th  of 
the  whole  length  of  its  stroke.  Let  the  stroke  of  the  piston  be  6 
feet,  or  72  inches.  We  wish  to  know  when  the  exhausting-port 
before  the  piston  will  be  closed,  and  when  the  one  behind  it  will  be 
opened.  At  the  top  of  the  left-hand  double  column,  the  given  de- 
gree of  expansion  (l-3d)  is  marked,  and  in  the  extreme  left  column 
we  have  at  the  top  the  given  amount  of  cover  (l-8th).  Opposite  the 
l-8th,  in  the  first  double  column,  we  have  "178  and  -033,  which 
decimals,  multiplied  respectively  by  72,  the  length  of  the  stroke, 
will  give  the  required  positions  of  the  piston:  thus  72x-178=12-8 
inches  =  distance  of  the  piston  from  the  end  of  the  stroke  when  the 
exhausting-port  before  the  piston  is  shut ;  and  72  x  -033  =  2-38 
inches  =  distance  of  the  piston  from  the  end  of  its  stroke  when  the 
exhausting-port  behind  it  is  opened. 


THE   STEAM   ENGINE. 


231 


0 

CO 

to 

c- 

H* 

1—  I 

O 

& 

oo 

Cover  on  the  exhausting  side  of  the  valve  in  parts  of 
the  length  of  its  stroke. 

s 

to 

»lt 
M 
CO 

HJ 

§ 

(-li 

2 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

00 

v-     S" 

CD    H-  >  P 

}&B 
fjfi 

*    g. 

i 

i 

i 

o 

g 

CO 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

g 

to 

o 

M 

H-l 

1—  » 
00 

OS 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

Steam*  cut  off  at 
7-24ths  from 
the  end  of  the 
stroke. 

i 

§ 

B 

o 

C71 

to 

i 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

i 

1 

i 

l-li 

t— 

CO 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

<r*          & 

r>-i 

TO     0     rf^    B 

!§.&§ 

r|!l 

0.          p 

i 

i 

2 

o 

0 
CD 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

i 

CD 

i 

l-li 

to 

O5 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

Steam  cut  off  at 
5-24ths  from 
the  end  of  the 
stroke. 

o 

Gi 
Gl 

O 
K 
K> 

o 

1 

0 

to 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

0 
£» 

CO 

0 
Ci 
CO 

0 
1—  » 

i 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

Steam  cut  off  at 
l-6th  from 
the  end  of  the 
stroke. 

i 

g 

CO 

g 

to 

o 

o 

CD 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

I 

CO 

o 

hU 
CO 

o 

C7X 
QO 

0 
SO 
CO 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

Steam  cut  off  at 
l-8th  from 
the  end  of  the 
stroke. 

o 

CO 
CO 

i 

0 
0» 

1 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  ofrthe  stroke). 

i 

o 

£ 

0 

— 

CO 

0 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke.) 

c*      w 
«  7"f 

£3£E 

Rr* 
r|p* 

0           g. 

o 

8 

2 

CO 

o 
o 

00 

§ 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

0 

|—» 

o 

£ 

o 
to 
-JI 

g 

CO 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  before 
it  is  shut  (in  parts  of  the  stroke). 

frf 

^is! 

lift 

:-3? 

0 
H-> 

4X 

i 

1 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting-port  behind 
it  is  opened  (in  parts  of  the  stroke). 

232  THE    PRACTICAL    MODEL    CALCULATOR. 

To  take  another  example.  Let  the  stroke  of  the  valve  be  16 
inches,  the  cover  on  the  exhausting  side  J  inch,  the  cover  on  the 
steam  side"3J  inches,  the  length  of  the  stroke  of  the  piston  60  inches. 
It  is  required  to  ascertain  all  the  particulars  of  the  working  of  this 
valve.  The  cover  on  the  exhausting  side  is  evidently  ¥12  of  the 
length  of  the  valve  stroke.  Again,  looking  at  16  in  the  left-hand 
column  of  Table  II.,  we  find  in  the  same  horizontal  line  3*26,  or  very 
nearly  3J  under  |  at  the  head  of  the  column,  thus  showing  that  the 
steam  will  be  cut  off  at  $  from  the  end  of  the  stroke.  Again,  under 
i  at  the  head  of  the  fifth  double  column  from  the  left  in  Table  III., 
and  in  a  horizontal  line  with  ^  in  the  left-hand  column,  we  have 
•053  and  -033.  Hence,  -053  X  60  =  3-18  inches  =  distance  of  the 
piston  from  the  end  of  its  stroke  when  the  exhausting-port  before 
it  is  shut,  and  -033  X  60  =  1-98  inches  =  distance  of  the  piston 
from  the  end  of  its  stroke  when  the  exhausting-port  behind  it  is 
opened.  If  in  this  valve  the  cover  on  the  exhausting  side  were 
increased  (say  to  2  inches,  or  $  of  the  stroke,)  the  effect  would  be  to 
make  the  port  before  the  valve  be  shut  sooner  in  the  proportion  of 
•109  to  *053,  and  the  port  behind  it  later  in  the  proportion  of  *008 
to  *033  (see  Table  III.)  Whereas,  if  the  cover  on  the  exhausting 
side  were  removed  entirely,  the  port  before  the  piston  would  be 
shut  and  that  behind  it  opened  at  the  same  time,  and  (see  bottom 
of  fifth  double  column,  Table  III.)  the  distance  of  the  piston  from 
the  end  of  its  stroke  at  that  time  would  be  *043  X  60  =  2'58  inches. 

An  inspection  of  Table  III.  shows  us  the  effect  of  increasing  the 
expansion  by  the  slide-valve  in  augmenting  the  loss  of  power  occa- 
sioned by  the  imperfect  action  of  the  eduction  passages.  Referring 
to  the  bottom  line  of  the  table,  we  see  that  the  eduction  passage 
before  the  piston  is  closed,  and  that  behind  it  opened,  (thus  destroy- 
ing the  whole  moving  power  of  the  engine,)  when  the  piston  is  '092 
from  the  end  of  its  stroke,  the  steam  being  cut  off  at  J  from  the 
end.  Whereas,  if  the  steam  is  only  cut  off  at  ^  from  the  end  of 
the  stroke,  the  moving  power  is  not  withdrawn  till  only  -Oil  of  the 
stroke  remains  uncompleted.  It  will  also  be  observed  that  in- 
creasing the  cover  on  the  exhausting  side  has  the  effect  of  retaining 
the  action  of  the  steam  longer  behind  the  piston,  but  it  at  the  same 
time  causes  the  eduction-port  before  it  to  be  closed  sooner. 

A  very  cursory  examination  of  the  action  of  the  slide  valve  is 
sufficient  to  show  that  the  cover  on  the  steam  side  should  always  be 
greater  than  on  the  exhausting  side.  If  they  are  equal,  the  steam 
would  be  admitted  on  one  side  of  the  piston  at  the  same  time  that 
it  was  allowed  to  escape  from  the  other ;  but  universal  experience 
has  shown  that  when  this  is  the  case,  a  very  considerable  part  of 
the  power  of  the  engine  is  destroyed  by  the  resistance  opposed  to 
the  piston,  by  the  exhausting  steam  not  getting  away  to  the  con- 
denser with  sufficient  rapidity.  Hence  we  see  the  necessity  of 
the  cover  on  the  exhausting  side  being  always  less  than  the  cover  on 
the  steam  side ;  and  the  difference  should  be  the  greater  the  higher 
the  velocity  of  the  piston  is  intended  to  be,  because  the  quicker  the 


THE    STEAM   ENGINE. 


233 


piston  moves  the  passage  for  the  waste  steam  requires  to  be  the 
larger,  so  as  to  admit  of  its  getting  away  to  the  condenser  with  as 
great  rapidity  as  possible.  In  locomotive  or  other  engines,  where 
it  is  not  wished  to  expand  the  steam  in  the  cylinder  at  all,  the  slide 
valve  is  sometimes  made  with  very  little  cover  on  the  steam  side : 
and  in  these  circumstances,  in  order  to  get  a  sufficient  difference 
between  the  cover  on  the  steam  and  exhausting  sides  of  the  valve, 
it  may  be  necessary  not  only  to  take  away  all  the  cover  on  the 
exhausting  side,  but  to  take  off  still  more,  so  as  to  make  both  ex- 
hausting passages  be  in  some  degree  open,  when  the  valve  is  at  the 
middle  of  its  stroke.  This,  accordingly,  is  sometimes  done  in  such 
circumstances 'as  we  have  described ;  but,  when  there  is  even  a  small 
degree  of  cover  on  the  steam  side,  this  plan  of  taking  more  than  all 
the  cover  off  the  exhausting  side  ought  never  to  be  resorted  to,  as 
it  can  serve  no  good  purpose,  and  will  materially  increase  an  evil 
we  have  already  explained,  viz.  the  opening  of  the  exhausting-port 
behind  the  piston  before  the  stroke  is  nearly  completed.  The  tables 
apply  equally  to  the  common  short  slide  three-ported  valves  and  to 
the  long  D  valves. 

In  fig.  1  is  exhibited  a  common  arrangement  of  the  valves  in  10. 


234  THE   PRACTICAL   MODEL   CALCULATOR. 

comotive  engines,  and  in  figs.  2  and  3  is  shown  an  arrangement 
for  working  valves  by  a  shifting  cam,  by  which  the  amount  of  ex- 
pansion may  be  varied.  This  particular  arrangement,  however,  is 
antiquated,  and  is  now  but  little  used. 

The  extent  to  which  expansion  can  be  carried  beneficially  by 
means  of  lap  upon  the  valve  is  about  one-third  of  the  stroke ;  that 
is,  the  valve  may  be  made  with  so  much  lap,  that  the  steam  will  be 
cut  off  when  one-third  of  the  stroke  has  been  performed,  leaving 
the  residue  to  be  accomplished  by  the  agency  of  the  expanding 
steam ;  but  if  more  lap  be  put  on  than  answers  to  this  amount  of 
expansion,  a  very  distorted  action  of  the  valve  will  be  produced, 
which  will  impair  the  efficiency  of  the  engine.  If  a  further  amount 
of  expansion  than  this  is  wanted,  it  may  be  accomplished  by  wire- 
drawing the  steam,  or  by  so  contracting  the  steam  passage,  that 
the  pressure  within  the  cylinder  must  decline  when  the  speed  of 
the  piston  is  accelerated,  as  it  is  about  the  middle  of  the  stroke. 
Thus,  for  example,  if  the  valve  be  so  made  as  to  shut  off  the  steam 
by  the  time  two-thirds  of  the  stroke  have  been  performed,  and  the 
steam  be  at  the  same  time  throttled  in  the  steam  pipe,  the  full 
pressure  of  the  steam  within  the  cylinder  cannot  be  maintained  ex- 
cept near  the  beginning  of  the  stroke  where  the  piston  travels 
slowly  ;  for  as  the  speed  of  the  piston  increases,  the  pressure  neces- 
sarily subsides,  until  the  piston  approaches  the  other  end  of  the 
cylinder,  where  the  pressure  would  rise  again  but  that  the  operation 
of  the  lap  on  the  valve  by  this  time  has  had  the  effect  of  closing 
the  communication  between  the  cylinder  and  steam  pipe,  so  as  to 
prevent  more  steam  from  entering.  By  throttling  the  steam,  there- 
fore, in  the  manner  here  indicated,  the  amount  of  expansion  due  to 
the  lap  may  be  doubled,  so  that  an  engine  with  lap  enough  upon 
the  valve  to  cut  off  the  steam  at  two-thirds  of  the  stroke,  may,  by 
the  aid  of  wire-drawing,  be  virtually  rendered  capable  of  cutting 
off  the  steam  at  one-third  of  the  stroke.  The  usual  manner  of  cut- 
ting off  the  steam,  however,  is  by  means  of  a  separate  valve,  termed 
an  expansion  valve  ;  but  such  a  device  appears  to  be  hardly  neces- 
sary in  many  engines.  In  the  Cornish  engines,  where  the  steam 
is  cut  off  in  some  cases  at  one-twelfth  of  the  stroke,  a  separate  valve 
for  the  admission  of  steam,  other  than  that  which  permits  its  es- 
cape, is  of  course  indispensable ;  but  in  common  rotative  engines, 
which  may  realize  expansive  efficacy  by  throttling,  a  separate  ex- 
pansive valve  does  not  appear  to  be  required.  In  all  engines  there 
is  a  point  beyond  which  expansion  cannot  be  carried  with  advantage, 
as  the  resistance  to  be  surmounted  by  the  engine  will  then  become 
equal  to  the  impelling  power ;  but  in  engines  working  with  a  high 
pressure  of  steam  that  point  is  not  so  speedily  attained. 

In  high  pressure,  as  contrasted  with  condensing  engines,  there  is 
always  the  loss  of  the  vacuum,  which  will  generally  amount  to  12 
or  13  Ibs.  on  the  square  inch,  and  in  high  pressure  engines  there  is 
a  benefit  arising  from  the  use  of  a  very  high  pressure  over  a  pres- 
sure of  a  moderate  account.  In  all  high  pressure  engines,  there  is 


THE    STEAM    ENGINE.  235 

a  diminution  in  the  power  caused  by  the  counteracting  pressure  of 
the  atmosphere  on  the  educting  side  of  the  piston  ;  for  the  force 
of  the  piston  in  its  descent  would  obviously  be  greater,  if  there  was 
a  vacuum  beneath  it ;  and  the  counteracting  pressure  of  the  atmo- 
sphere is  relatively  less  when  the  steam  used  is  of  a  very  high 
pressure.  It  is  clear,  that  if  we  bring  down  the  pressure  of  the 
steam  in  a  high  pressure  engine  to  the  pressure  of  the  atmosphere, 
it  will  not  exert  any  power  at  all,  whatever  quantity  of  steam  may 
be  expended,  and  if  the  pressure  be  brought  nearly  as  low  as  that 
of  the  atmosphere,  the  engine  will  exert  only  a  very  small  amount 
of  power  ;  whereas,  if  a  very  high  pressure  be  employed,  the  pres- 
sure of  the  atmosphere  will  become  relatively  as  small  in  counter- 
acting the  impelling  pressure,  as  the  attenuated  vapour  in  the  con- 
denser of  a  condensing  engine  is  in  resisting  the  lower  pressure 
which  is  there  employed.  Setting  aside  loss  from  friction,  and  sup- 
posing the  vacuum  to  be  a  perfect  one,  there  would  be  no  benefit 
arising  from  the  use  of  steam  of  a  high  pressure  in  condensing  en- 
gines, for  the  same  weight  of  steam  used  without  expansion,  or 
with  the  same  measure  of  expansion,  would  produce  at  every  pres- 
sure the  same  amount  of  mechanical  power.  A  piston  with  a 
square  foot  of  area,  and  a  stroke  of  three  feet  with  a  pressure  of 
one  atmosphere,  would  obviously  lift  the  same  weight  through  the 
same  distance,  as  a  cylinder  with  half  a  square  foot  of  area,  a  stroke 
of  three  feet,  and  a  pressure  of  two  atmospheres.  In  the  one  case, 
we  have  three  cubic  feet  of  steam  of  the  pressure  of  one  atmosphere, 
and  in  the  other  case  1J  cubic  feet  of  the  pressure  of  two  atmo- 
spheres. But  there  is  the  same  weight  of  steam,  or  the  same  quan- 
tity of  heat  and  water  in  it,  in  both  cases  ;  so  that  it  appears  a  given 
weight  of  steam  would,  under  such  circumstances,  produce  a  definite 
amount  of  power,  without  reference  to  the  pressure.  In  the  case 
of  ordinary  engines,  however,  these  conditions  do  not  exactly  apply ; 
the  vacuum  is  not  a  perfect  one,  and  the  pressure  of  the  resisting 
vapour  becomes  relatively  greater  as  the  pressure  of  the  steam  is 
diminished ;  the  friction  also  becomes  greater  from  the  necessity 
of  employing  larger  cylinders,  so  that  even  in  the  case  of  condensing 
engines,  there  is  a  benefit  arising  from  the  use  of  steam  of  a  con- 
siderable pressure.  Expansion  cannot  be  carried  beneficially  to  any 
great  extent,  unless  the  initial  pressure  be  considerable;  for  if  steam 
of  a  low  pressure  were  used,  the  ultimate  tension  would  be  reduced 
to  a  point  so  nearly  approaching  that  of  the  vapour  in  the  con- 
denser, that  the  difference  would  not  suffice  to  overcome  the  friction 
of  the  piston  ;  and  a  loss  of  power  would  be  occasioned  by  carrying 
expansion  to  such  an  extent.  In  some  of  the  Cornish  engines,  the 
steam  is  cut  off  at  one-twelfth  of  the  stroke  ;  but  there  would  be  a 
loss  arising  from  carrying  the  expansion  so  far,  instead  of  a  gain, 
unless  the  pressure  of  the  steam  were  considerable.  It  is  clear, 
that  in  the  case  of  engines  which  carry -expansion  very  far,  a  very 
perfect  vacuum  in  the  condenser  is  more  important  than  it  is  in 
other  cases.  Nothing  can  be  easier  than  to  compute  the  ultimate 


236          THE  PRACTICAL  MODEL  CALCULATOR. 

pressure  of  expanded  steam,  so  as  to  see  at  what  point  expansion 
ceases  to  be  productive  of  benefit ;  for  as  the  pressure  of  expanded 
steam  is  inversely  as  the  space  occupied,  the  terminal  pressure  when 
the  expansion  is  twelve  times  is  just  one-twelfth  of  what  it  was  at 
first,  and  so  on,  in  all  other  projections.  The  total  pressure  should 
be  taken  as  the  initial  pressure — not  the  pressure  on  the  safety 
valve,  but  that  pressure  plus  the  pressure  of  the  atmosphere. 

In  high  pressure  engines,  working  at  from  70  to  90  Ibs.  on  the 
square  inch,  as  in  the  case  of  locomotives,  the  efficiency  of  a  given 
quantity  of  water  raised  into  steam  may  be  considered  to  be  about 
the  same  as  in  condensing  engines.  If  the  pressure  of  steam  in  a 
high  pressure  engine  be  120  Ibs.,  or  125  Ibs.  above  the  atmosphere, 
then  the  resistance  occasioned  by  the  atmosphere  will  cause  a  loss 
of  |th  of  the  power.  If  the  pressure  of  the  steam  in  a  low  pressure 
engine  be  16  Ibs.  on  the  square  inch,  or  11  Ibs.  above  the  atmo- 
sphere, and  the  tension  of  the  vapour  in  the  condenser  be  equiva- 
lent to  4  inches  of  mercury,  or  2  Ibs.  of  pressure  on  the  square 
inch,  then  the  resistance  occasioned  by  this  rare  vapour  will  also 
cause  a  loss  of  fth  of  the  power.  A  high  pressure  engine,  there- 
fore, with  a  pressure  of  105  Ibs.  above  the  atmosphere,  works  with 
only  the  same  loss  from  resistance  to  the  piston,  as  a  low  pressure 
engine  with  a  pressure  of  1  Ib.  above  the  atmosphere,  and  with 
these  proportions  the  power  produced  by  a  given  weight  of  steam 
will  be  the  same,  whether  the  engine  be  high  pressure  or  con- 
densing. 

SPHEROIDAL   CONDITION    OP   WATER   IN   BOILERS. 

Some  of  the  more  prominent  causes  of  boiler  explosions  have 
been  already  enumerated;  but  explosions  have  in  some  cases  been 
attributed  to  the  spheroidal  condition  of  the  water  in  the  boiler, 
consequent  upon  the  flues  becoming  red-hot  from  a  deficiency  of 
water,  the  accumulation  of  scale,  or  otherwise.  The  attachment 
of  scale,  from  its  imperfect  conducting  power,  will  cause  the  iron 
to  be  unduly  heated ;  and  if  the  scale  be  accidentally  detached,  a 
partial  explosion  may  occur  in  consequence.  It  is  found,  that  a 
sudden  disengagement  of  steam  does  not  immediately  follow  the 
contact  of  water  with  the  hot  metal,  for  water  thrown  upon  red- 
hot  iron  is  not  immediately  converted  into  steam,  but  assumes  the 
spheroidal  form  and  rolls  about  in  globules  over  the  surface.  These 
globules,  however  high  the  temperature  of  the  metal  may  be  on 
which  they  are  placed,  never  rise  above  the  temperature  of  205°, 
and  give  off  but  very  little  steam  ;  but  if  the  temperature  of  the 
metal  be  lowered,  the  water  ceases  to  retain  the  spheroidal  form, 
and  comes  into  intimate  contact  wTith  the  metal,  whereby  a  rapid 
disengagement  of  steam  takes  place.  If  water  be  poured  into  a  very 
hot  copper  flask,  the  flask  may  be  corked  up,  as  there  will  be  scarce 
any  steam  produced  so  long  as  the  high  temperature  is  maintained ; 
but  so  soon  as  the  temperature  is  suffered  to  fall  below  350°  or 
400°,  the  spheroidal  condition  being  no  longer  maintainable,  steam 
is  generated  with  rapidity,  and  the  cork  will  be  projected  from  the 


THE    STEAM    ENGINE.  237 

mouth  of  the  flask  with  great  force.  In  a  boiler,  no  doubt,  where 
there  is  a  considerable  head  of  water,  the  repellant  action  of  the 
spheroidal  globules  will  be  more  effectually  counteracted  than  in 
the  small  vessels  employed  in  experimental  researches.  But  it  is 
doubtful  whether  in  all  boilers  there  may  not  be  something  of  the 
spheroidal  action  perpetually  in  operation,  and  leading  to  effects  at 
present  mysterious  or  inexplicable. 

One  of  the  most  singular  phenomena  attending  the  spheroidal 
condition  is,  that  the  vapour  arising  from  a  spheroid  is  of  a  far 
higher  temperature  than  the  spheroid  itself.  Thus,  if  a  thermometer 
be  held  in  the  atmosphere  of  vapour  which  surrounds  a  spheroid  of 
water,  the  mercury,  instead  of  standing  at  205°,  as  would  be  the 
case  if  it  had  been  immersed  in  the  spheroid,  will  rise  to  a  point 
determinable  by  the  temperature  of  the  vessel  in  which  the  spheroid 
exists.  In  the  case  of  a  spheroid,  for  example,  existing  within  a 
crucible  raised  to  a  temperature  of  400°,  the  thermometer,  if  held 
in  the  vapour,  will  rise  to  that  point ;  and  if  the  crucible  be  made 
red-hot,  the  thermometer  will  be  burst,  from  the  boiling  point  of 
mercury  having  been  exceeded.  A  part  of  this  effect  may,  indeed, 
be  traced  to  direct  radiation,  yet  it  appears  indisputable,  from  the 
experiments  which  have  been  made,  that  the  vapour  of  a  liquid 
spheroid  is  much  hotter  than  the  spheroid  itself. 

EXPANSION. 

At  page  131  we  have  given  a  table  of  hyperbolic  or  Byrgean 
logarithms,  for  the  purpose  of  facilitating  computations  upon  this 
subject. 

Let  the  pressure  of  the  steam  in  the  boiler  be  expressed  by  unity, 
and  let  x  represent  the  space  through  which  the  piston  has  moved 
whilst  urged  by  the  expanding  steam.  The  density  will  then  be 

— -,  and.  assuming  that  the  densities  and  elasticities  are  pro- 

1  +  x 

portionate,  - — ?—  will  be  the  differential  of  the  efficiency,  and  the 
1  4-  x 

efficiency  itself  will  be  the  integral  of  this,  or,  in  other  words,  the 
hyperbolic  logarithm  of  the  denominator ;  wherefore  the  efficiency 
of  the  whole  stroke  will  be  1  +  log.  (1  -f  x). 

Supposing  the  pressure  of  the  atmosphere  to  be  15  Ibs.,  15  +  35 
=  50  Ibs.,  and  if  the  steam  be  cut  off  at  Jth  of  the  stroke,  it  will  be 
expanded  into  four  times  its  original  volume ;  so*  that  at  the  ter- 
mination of  the  stroke,  its  pressure  will  be  50-r-4=12-2  Ibs.,  or  2-8 
Ibs.  less  than  the  atmospheric  pressure. 

When  the  steam  is  cut  off  at  one-fourth,  it  is  evident  that  x  =  3. 
In  such  case  the  efficiency  is 

1  -f  log.  (1  +  3),  or  1  +  log.  4. 

The  hyperbolic  logarithm  of  4  is  1-386294,  so  that  the  efficiency 
of  the  steam  becomes  2-386294 ;  that  is,  by  cutting  off  the  steam 
at  J,  more  than  twice  the  effect  is  produced  with  the  same  consump- 
tion of  fuel ;  in  other  words,  one-half  of  the  fuel  is  saved. 


238  THE    PRACTICAL   MODEL   CALCULATOR. 

This  result  may  thus  be  expressed  in  words  : — Divide  the  length 
of  the  stroke  through  which  the  steam  expands  by  the  length  of 
stroke  performed  with  the  full  pressure,  which  last  portion  call  1 ; 
the  hyperbolic  logarithm  of  the  quotient  is  the  increase  of  efficiency 
due  to  expansion.  We  introduce  on  the  following  page  more  de- 
tailed tables,  to  facilitate  the  computation  of  the  power  of  an  en- 
gine working  expansively,  or  rather  to  supersede  the  necessity  of 
entering  into  a  computation  at  all  in  each  particular  case. 

The  first  column  in  each  of  the  following  tables  contains  the 
initial  pressure  of  the  steam  in  pounds,  and  the  remaining  columns 
contain  the  mean  pressure  of  steam  throughout  the  stroke,  with  the 
different  degrees  of  expansion  indicated  at  the  top  of  the  columns, 
and  which  express  the  portion  of  the  stroke  during  which  the  steam 
acts  expansively.  Thus,  for  example,  if  steam  be  admitted  to  the 
cylinder  at  a  pressure  of  3  pounds  per  square  inch,  and  be  cut  oif 
within  Jth  of  the  end  of  the  stroke,  the  mean  pressure  during  the 
whole  stroke  will  be  2*96  pounds  per  square  inch.  In  like  manner, 
if  steam  at  the  pressure  of  3  pounds  per  square  inch  were  cut  off 
after  the  piston  had  gone  through  Jth  of  the  stroke,  leaving  the 
steam  to  expand  through  the  remaining  Jth,  the  mean  pressure 
during  the  whole  stroke  would  be  1*164  pounds  per  square  inch. 

FRICTION. 

The  friction  of  iron  sliding  upon  brass,  which  has  been  oiled  and 
then  wiped  dry,  so  that  no  film  of  oil  is  interposed,  is  about  ^  of 
the  pressure ;  but  in  machines  in  actual  operation,  where  there  is  a 
film  of  oil  between  the  rubbing  surfaces,  the  fraction  is  only  about 
one-third  of  this  amount,  or  ^d  of  the  weight.  The  tractive  re- 
sistance of  locomotives  at  low  speeds,  which  is  entirely  made  up  of 
friction,  is  in  some  cases  ^th  of  the  weight ;  but  on  the  average 
about  ^th  of  the  load,  which  nearly  agrees  with  my  former  state- 
ment. If  the  total  friction  be  ^th  of  the  load,  and  the  rolling 
friction  be  y^th  of  the  load,  then  the  friction  of  attrition  must  be 
?|gth  of  the  load ;  and  if  the  diameter  of  the  wheels  be  36  in.,  and  the 
diameter  of  the  axles  be  3  in.,  which  are  common  proportions,  the 
friction  of  attrition  must  be  increased  in  the  proportion  of  36  to  3, 
or  12  times,  to  represent  the  friction  of  the  rubbing  surface  when 
moving  with  the  velocity  of  the  carriage.  ?%ths  are  about  ^th  of 
the  load,  which  does  not  differ  much  from  the  proportion  of  g^d,  as 
previously  stated.  While  this,  however,  is  the  average  result,  the 
friction  is  a  good  deal  less  in  some  cases.  Engineers,  in  some 
experiments  upon  the  friction,  found  the  friction  to  amount  to 
less  than  ^th  of  the  weight ;  and  in  some  experiments  upon  the 
friction  of  locomotive  axles,  it  was  found  that  by  ample  lubrication 
the  friction  might  be  made  as  little  as  ^th  of  the  weight,  and  the 
traction,  with  the  ordinary  size  of  wheels,  would  in  such  a  case  be 
about  5^th  of  the  weight.  The  function  of  lubricating  substances 
is  to  prevent  the  rubbing  surfaces  from  coming  into  contact,  where- 
by abrasion  would  be  produced,  and  unguents  are  effectual  in  this 


THE    STEAM    ENGINE. 


239 


EXPANDED    STEAM. — MEAN    PRESSURE   AT   DIFFERENT   DENSITIES   AND 
RATE   OF   EXPANSION. 

The  column  headed  0  contains  the  initial  pressure  in  Ibs.,  and  the  remaining  columns 
contain  the  mean  pressure  in  Ibs.,  with  different  grades  of  expansion. 


EXPANSION  BY  EIGHTHS. 

0 

1 

8 

f 

1 

f 

! 

7 
I 

3 

2-96 

2-89 

2-75 

2-53 

2-22 

1-789 

1-154 

4 

3-95 

3-85 

3-67 

3-38 

2-96 

2-386 

1-539 

5 

4-948 

4-818 

4-593 

4-232 

3-708 

2-982 

1-924 

6 

6-937 

5-782 

5-512 

5-079 

4-450 

3-579 

2-309 

7 

6-927 

6-746 

6-43L 

5-925 

5-241 

4-175 

2-694 

8 

7-917 

7-710 

7-350 

6-772 

5-934 

4-772 

3-079 

9 

8-906 

8-673 

8-268 

7-618 

6-675 

5-368 

3-463 

10 

9-896 

9-637 

9-187 

8-465 

7-417 

5-965 

3-848 

11 

10-885 

10-601 

10-106 

9-311 

8-159 

6-561 

4-233 

12 

11-875 

11-565 

10-925 

10-158 

8-901 

7-158 

4-618 

13 

12-865 

12-528 

11-943 

11-004 

9-642 

7-754 

5-003 

14 

13-854 

13-492 

12-862 

11-851 

10-384 

8-531 

5-388 

15 

14-844 

14-456 

13-781 

12-697 

11-126 

8-947 

5-773 

16 

15-834 

15-420 

14-700 

13-544 

11-868 

9-544 

6-158 

17 

16-823 

16-383 

15-618 

14-390 

12-609 

10-140 

6-542 

18 

17-813 

17-347 

16-537 

15-237 

13-351 

10-737 

6-927 

19 

18-702 

18-311 

17-448 

16-803 

14-093 

11-333 

7-312 

20 

19-792 

19-275 

18-375 

16-930 

14-835 

11-930 

7-697 

25 

24-740 

24-093 

22-968 

21-162 

18-543 

14-912 

9-621 

30 

29-688 

28-912 

27-562 

25-395 

22-252 

17-895 

11-546 

35 

34-636 

33-731 

33-156 

29-627 

25-961 

20-877 

13-470 

40 

39-585 

38-550 

36-750 

33-860 

29-670 

23-860 

15-395 

45 

44-533 

43-368 

41-343 

38-092 

33-378 

26-842 

17-319 

50 

49-481 

48-187 

45-937 

42-325 

37-067 

29-825 

19-243 

EXPANSION  BY  TENTHS. 

0 

A 

JL 

10 

lo            I4o 

I6o 

& 

I7o 

A 

A 

3 

2-980 

2-930 

2-830 

2-710 

2-539 

2-299 

1-981 

1-668 

0-990 

4 

3-974 

3-913 

3-780 

3-614 

3-386 

3-065 

2-642 

2-087 

1-320 

5 

4-968 

4-892 

4-725 

4-518 

4-232 

3-832 

3-303 

2-609 

1-651 

6 

5-961 

5-870 

5-670 

5-421 

5-079 

4-598 

3-963 

3-130 

1-981 

7 

6-955 

6-848 

6-615 

6-325 

5-925 

5-364 

4-624 

3-652 

2-311 

8 

7-948 

7-827 

7-560 

7-228 

6-772 

6-131 

6-284 

4-174 

2-641 

9 

8-942 

8-805 

8-505 

8-132 

7-618 

6-897 

6-945 

4-696 

2-971 

10 

9-936 

9-784 

9-450 

9-036 

8-465 

7-664 

6-606 

5-218 

3-302 

11 

10-929 

10-762 

10-395 

9-939 

9-311 

8-430 

7-266 

5-739 

3-632 

12 

11-923 

11-740 

11-340 

10-843 

10-158 

9-196 

7-927 

6-261 

3-962 

13 

12-856 

12-719 

12-285 

11-746 

10-994 

9-963 

8-587 

6-783 

4-292 

14 

13-910 

13-967 

13-230 

12-650 

11-851 

10-729 

9-248 

7-305 

4-622 

15 

14-904 

14-676 

14-175 

13-554 

12-697 

11-496 

9-909 

7-827 

4-953 

16 

15-897 

15-654 

15-120 

14-457 

13-544 

12-262    10-569 

8-348 

5-283 

17 

16-891 

16-632 

16-065 

15-361 

14-051 

13-028 

11-230 

8-870 

5-613 

18 

17-884 

17-611 

17-010 

16-264 

15-237 

13-795 

11-890 

9-392 

5-944 

19 

18-878 

18-589 

17-955 

17-168 

16-083 

14-561 

12-551 

9-914 

6-273 

20 

19-872 

19-568 

18-900 

18-072 

16-930 

15-328 

13-212 

10-436 

6-600 

25 

24-840 

24-460 

23-625 

22-590 

21-162 

19-160 

16-515 

13-040 

8-255 

30 

29-808 

29-352 

28-350 

27-108 

25-395 

22-992 

19-818 

15-654 

9-906 

35 

34-776 

34-244 

33-075 

31-626 

29-627 

26-824 

23-121 

18-263 

11-557 

40 

39-744 

39-136 

37-800 

36-144 

33-860 

30-656 

26-224 

20-872 

13-208 

45  :  44-912 

44-028 

42-525 

40-662 

38-092 

34-888 

29-727 

23-481 

14-859 

50    49-680 

48-920 

47-250 

45-180 

42-325 

38-320 

33-030 

26-090 

16-510 

240  THE    PRACTICAL   MODEL    CALCULATOR. 

respect  in  the  proportion  of  their  viscidity ;  but  if  the  viscidity  of 
the  unguent  be  greater  than  what  suffices  to  keep  the  surfaces 
asunder,  an  additional  resistance  will  be  occasioned  ;  and  the  nature 
of  the  unguent  selected  should  always  have  reference,  therefore,  to 
the  size  of  the  rubbing  surfaces,  or  to  the  pressure  per  square  inch 
upon  them.  With  oil,  the  friction  appears  to  be  a  minimum  when 
the  pressure  on  the  surface  of  a  bearing  is  about  90  Ibs.  per  square 
inch  :  the  friction  from  too  small  a  surface  increases  twice  as  rapidly 
as  the  friction  from  too  large  a  surface ;  added  to  which,  the  bear- 
ing, when  the  surface  is  too  small,  wears  rapidly  away.  For  all 
sorts  of  machinery,  the  oil  of  Patrick.  Sarsfield  Devlan,  of  Reading, 
Pa.,  is  the  best. 

HORSE   POWER. 

A  horse  power  is  an  amount  of  mechanical  force  capable  of  rais- 
ing 33,000  Ibs.  one  foot  high  in  a  minute.  The  average  force  ex- 
erted by  the  strongest  horses,  amounting  to  33,000  Ibs.,  raised  one 
foot  high  in  the  minute,  was  adopted,  and  has  since  been  retained. 
The  efficacy  of  engines  of  a  given  size,  however,  has  been  so  much 
increased,  that  the  dimensions  answerable  to  a  horse  power  then, 
will  raise  much  more  than  33,000  Ibs.  one  foot  high  in  the  minute 
now ;  so  that  an  actual  horse  power,  and  a  nominal  horse  power 
are  no  longer  convertible  terms.  In  some  engines  every  nominal 
horse  power  will  raise  52,000  Ibs.  one  foot  high  in  the  minute,  in 
others  60,000  Ibs.,  and  in  others  66,000  Ibs. ;  so  that  an  actual  and 
nominal  horse  power  are  no  longer  comparable  quantities, — the  one 
being  a  unit  of  dimension,  and  the  other  a  unit  of  force.  The  ac- 
tual horse  power  of  an  engine  is  ascertained  by  an  instrument  called 
an  indicator ;  but  the  nominal  power  is  ascertained  by  a  reference 
to  the  dimensions  of  the  cylinder,  and  may  be  computed  by  the 
following  rule : — Multiply  the  square  of  the  diameter  of  the  cylin- 
der in  inches  by  the  velocity  of  the  piston  in  feet  per  minute,  and 
divide  the  product  by  6,000  ;  the  quotient  is  the  number  of  nominal 
horses  power.  In  using  this  rule,  however,  it  is  necessary  to  adopt 
the  speed  of  piston  which  varies  with  the  length  of  the  stroke.  The 
speed  of  piston  with  a  two  feet  stroke  is,  according  to  this  system, 
160  per  minute  ;  with  a  2  ft.  6  in.  stroke,  170  ;  3  ft.,  180  ;  3  ft.,  6 
in.,  189  ;  4  ft.,  200  ;  5  ft.,  215  ;  6  ft.,  228  ;  7  ft.,  245  ;  8  ft.,  256  ft. 

By  ascertaining  the  ratio  in  which  the  velocity  of  the  piston 
increases  with  the  length  of  the  stroke,  the  element  of  velocity  may 
be  cast  out  altogether ;  and  this  for  most  purposes  is  the  most  con- 
venient method  of  procedure.  To  ascertain  the  nominal  power  by 
this  method,  multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  the  cube  root  of  the  stroke  in  feet,  and  divide  the  pro- 
duct by  47 ;  the  quotient  is  the  number  of  nominal  horses  power 
of  the  engine.  This  rule  supposes  a  uniform  eifective  pressure  upon 
'the  .  piston  of  7  Ibs.  per  square  inch  ;  the  eifective  pressure  upon 
the  piston  of  4  horse  power  engines  of  some  of  the  best  makers 
has  been  estimated  at  6*8  Ibs.  per  square  inch,  and  the  pressure 


THE    STEAM    ENGINE.  241 

increased  slightly  with  the  power,  and  became  6-94  Ibs.  per  square 
inch  in  engines  of  100  horse  power ;  but  it  appears  to  be  more  con- 
venient to  take  a  uniform  pressure  of  7  Ibs.  for  all  powers.  Small 
engines,  indeed,  are  somewhat  less  effective  in  proportion  than  large 
ones  ;  but  the  difference  can  be  made  up  by  slightly  increasing  the 
pressure  in  the  boiler ;  and  small  boilers  will  bear  such  an  increase 
without  inconvenience. 

Nominal  power,  it  is  clear,  cannot  be  transformed  into  actual 
power,  for  the  nominal  horse  power  expresses  the  size  of  an  engine, 
and  the  actual  horse  power  the  number  of  times  33,000  Ibs.  it  will 
lift  one  foot  high  in  a  minute.  To  find  the  number  of  times  33,000 
Ibs.  or  528  cubic  feet  of  water,  an  engine  will  raise  one  foot  high 
in  a  minute, — or,  in  other  words,  the  actual  power, — we  first  find  the 
pressure  in  the  cylinder  by  means  of  the  indicator,  from  which  we 
deduct  a  pound  and  a  half  of  pressure  for  friction,  the  loss  of 
power  in  working  the  air  pump,  &c. ;  multiply  the  area  of  the 
piston  in  square  inches  by  this  residual  pressure,  and  by  the  motion 
of  the  piston,  in  feet  per  minute,  and  divide  by  33,000 ;  the 
quotient  is  the  actual  number  of  horse  power.  The  same  result  is 
attained  by  squaring  the  diameter  of  -the  cylinder,  multiplying  by 
the  pressure  per  square  inch,  as  shown  by  the  indicator,  less  a  pound 
and  a  half,  and  by  the  motion  of  the  piston  in  feet,  and  dividing  by 
42,017.  The  quantity  thus  arrived  at,  will,  in  the  case  of  nearly  all 
modern  engines,  be  very  different  from  that  obtained  by  multiplying 
the  square  of  the  diameter  of  the  cylinder  by  the  cube  root  of  the 
stroke,  and  dividing  by  47,  which  expresses  the  nominal  power ;  and 
the  actual  and  nominal  power  must  by  no  means  be  confounded,  as 
they  are  totally  different  things.  The  duty  of  an  engine  is  the 
work  done  in  relation  to  the  fuel  consumed,  and  in  ordinary  mill  or 
marine  engines  it  can  only  be  ascertained  by  the  indicator,  as  the 
load  upon  such  engines  is  variable,  and  cannot  readily  be  deter- 
mined :  but  in  the  case  of  engines  for  pumping  water,  where  the 
load  is  constant,  the  number  of  strokes  performed  by  the  engine 
represents  the  duty ;  and  a  mechanism  to  register  the  number  of 
strokes  made  by  the  engine  in  a  given  time,  is  a  sufficient  test  of 
the  engine's  performance. 

In  high  pressure  engines  the  actual  power  is  readily  ascertained 
by  the  indicator,  by  the  same  process  by  which  the  actual  power  of 
low  pressure  engines  is  ascertained.  The  friction  of  a  locomotive 
engine  when  unloaded,  is  found  by  experiment  to  be  about  1  Ib.  per 
square  inch  on  the  surface  of  the  pistons,  and  the  additional  friction 
caused  by  any  additional  resistance  is  estimated  at  about  -14  of 
that  resistance ;  but.  it  will  be  a  sufficiently  near  approximation  to 
the  power  consumed  by  friction  in  high  pressure  engines,  if  we 
make  a  deduction  of  a  pound  and  a  half  from  the  pressure  on  that 
account,  as  in  the  case  of  low  pressure  engines.  High  pressure 
engines,  it  is  true,  have  no  air  pump  to  work ;  but  the  deduction  of 
a  pound  and  a  half  of  pressure  is  relatively  a  much  smaller  one 
where  the  pressure  is  high  than  where  it  does  not  much  exceed  the 
V  16 


242  THE   PRACTICAL   MODEL   CALCULATOR. 

pressure  of  the  atmosphere.  The  rule,  therefore,  for  the  actual 
horse  power  of  a  high  pressure  engine  will  stand  thus  : — Square 
the  diameter  of  the  cylinder  in  inches,  multiply  by  the  pressure  of 
the  steam  in  the  cylinder  per  square  inch,  less  1J  Ibs.,  and  by  the 
speed  of  the  piston  in  feet  per  minute,  and  divide  by  42,017  ;  the 
quotient  is  the  actual  horse  power.  The  nominal  horse  power  of  a 
high  pressure  engine  has  never  .  been  defined ;  but  it  should  obvi- 
ously hold  the  same  relation  to  the  actual  power  as  that  which 
obtains  in  the  case  of  condensing  engines,  so  that  an  engine  of  a 
given  nominal  horse  power  may  be  capable  of  performing  the  same 
work,  whether  high  pressure  or  condensing.  This  relation  is  main- 
tained in  the  following  rule,  which  expresses  the  nominal  horse 
power  of  high  pressure  engines  : — Multiply  the  square  of  the  diame- 
ter of  the  cylinder  in  inches  by  the  pressure  on  the  piston  in  pounds 
per  square  inch,  and  by  the  speed  of  the  piston  in  feet  per  minute, 
and  divide  the  product  by  120,000 ;  the  quotient  is  the  power  of 
the  engine  in  nominal  horses  power.  If  the  pressure  upon  the 
piston  be  80  Ibs.  per  square  inch,  the  operation  may  be  abbreviated 
by  multiplying  the  square  of  the  diameter  of  the  cylinder  by  the 
speed  of  the  piston,  and  dividing  by  1,500,  which  will  give  the 
same  result.  This  rule  for  nominal  horse  power,  however,  is  not 
representative  of  the  dimensions  of  the  cylinder  ;  but  a  rule  for  the 
nominal  horse  power  of  high  pressure  engines  which  shall  discard 
altogether  the  element  of  velocity,  is  easily  constructed  ;  and,  as 
different  pressures  are  used  in  different  engines,  the  pressure  must 
become  an  element  in  the  computation.  The  rule  for  the  nominal 
power  will  therefore  stand  thus : — Multiply  the  square  of  the 
diameter  of  the  cylinder  in  inches  by  the  pressure  on  the  piston  in 
pounds  per  square  inch,  and  the  cube  root  of  the  stroke  in  feet,  and 
divide  the  product  by  940  ;  the  quotient  is  the  power  of  the  engine 
in  nominal  horse  power,  the  engine  working  at  the  ordinary  speed 
of  128  times  the  cube  root  of  the  stroke. 

A  summary  of  the  results  arrived  at  by  these  rules  is  given  in 
the  following  tables,  which,  for  the  convenience  of  reference,  we 
introduce. 


PARALLEL   MOTION. 

RULE  I. — In  such  a  combination  of  two  levers  as  is  represented  in 
Figs.  1  and  2,  page  245,  to  find  the  length  of  radius  bar  required 
for  any  given  length  of  lever  0  6r,  and  proportion  of  parts  of  the  link, 
G-E  and  F  E,  so  as  to  make  the  point  E  move  in  a  perpendicular 
line. — Multiply  the  length  of  G  C  by  the  length  of  the  segment  G  E, 
and  divide  the  product  by  the  length  of  the  segment  F  E.  The 
quotient  is  the  length  of  the  radius  bar. 

RULE  II. — (Fig.  2,  page  245.)  The  length  of  the  radius  bar  and 
of  C  G-  being  given,  to  find  the  length  of  the  segment  (F E}  of  the 
link  next  the  radius  bar. — Multiply  the  length  of  C  G  by  the 


THE   STEAM   ENGINE. 


243 


TABLE  of  Nominal  Horse  Power  of  Loiv  Pressure  Engines. 


Diameter  of 
Cjlinder  in 

LENGTH  OF  STROKE  ra  FEET. 

1 

1* 

2 

2* 

3 

3£ 

4 

4* 

5 

5* 

6 

7 

4 

•34 

•39 

•43 

•46 

•49 

•52 

•54 

•56 

•58 

•60 

•62 

•65 

5 

•53 

•61 

•67 

•72 

•76 

•81 

•84 

•88 

•91 

•94 

•96 

1-02 

6 

•76 

•87 

•96 

1-04 

1-10 

1-16 

1-22 

1-26 

1-31 

1-35 

1-39 

1-47 

7 

1-04 

1-19 

1-31 

1-41 

1-50 

1-58 

1-65 

1-72 

1-78 

1-84 

1-89 

1-99 

8 

1-36 

1-56 

1-72 

1-85 

1-96 

2-T)7 

2-16 

2-25 

2-33 

2-40 

2-47 

2-60 

9 

1-72 

1-97 

2-17 

2-34 

2-49 

2-62 

2-74 

2-84 

2-95 

3-04 

3-13 

3-30 

10 

2-13 

2-44 

2-68 

2-89 

3-07 

3-23 

3-38 

3-51 

3-64 

3-76 

3-87 

4-07 

11 

2-57 

2-95 

3-24 

3-49 

3-77 

3-91 

4-15 

4-25 

4-40 

4-54 

4-68 

4-92 

12 

3-06 

3-51 

3-86 

4-16 

4-42 

4-65 

4-86 

5-06 

5-24 

5-41 

5-57 

5-86 

13 

3-60 

4-12 

4-53 

4-88 

5-19 

5-46 

5-64 

5-94 

6-15 

6-35 

6-53 

6-88 

14 

4-17 

4-77 

5-25 

5-66 

6-01 

6-33 

6-62 

6-88 

7-13 

7-36 

7-58 

7-98 

15 

4-77 

5-48 

6-03 

6-50 

6-90 

7-27 

7-60 

7-90 

8-19 

8-45 

8-70 

9-16 

16 

5-45 

6-23 

6-86 

7-39 

7-86 

8-27 

8-65 

8-99 

9-31 

9-61 

9-90 

10-42 

17 

6-15 

7-04 

7-75 

8-35 

8-86 

9-34 

9-76 

10-15 

10-52 

10-85 

11-17 

11-76 

18 

6-89 

7-89 

8-68 

9-36 

9-94 

10-47 

10-94 

11-38 

11-79 

12-17 

12-53 

13-19 

19 

7-68 

8-79 

9-68 

10-42 

11-17 

11-66 

12-19 

12-68 

13-13 

13-56 

13-96 

14-69 

20 

8-51 

9-74 

10-72 

11-55 

12-27 

12-92 

13-51 

14-05 

14-55 

.   15-02 

15-46 

16-28 

22 

10-30 

11-79 

12-97 

13-98 

14-85 

15-63 

16-62 

17-30 

17-65 

18-18 

18-71 

19-70 

24 

12-26 

14-03 

15-44 

16-63 

17-67 

18-61 

19-45 

20-23 

20-95 

31-63 

22-27 

23-44 

26 

14-39 

16-46 

18-12 

19-52 

20-75 

21-84 

22-56 

23-75 

24-6 

25-39 

26-14 

27-51 

28 

16-68 

19-09 

21-02 

22-64 

24-06 

25-33 

26-48 

27-54 

28-52 

29-44 

30-31 

31-90 

30 

19-15 

21-92 

24-13 

25-99 

27-62 

29-07 

30-40 

31-61 

32-74 

33-80 

34-80 

36-63 

32 

21-79 

24-96 

27-51 

29-57 

31-42 

33-08 

34-59 

35-97 

37-26 

38-46 

39-59 

41-68 

34 

24-60 

28-16 

30-99 

33-39 

35-44 

37-34 

39-04 

40-60 

42-06 

43-41 

44-69 

47-05 

36 

27-57 

31-56 

34-74 

37-42 

39-77 

41-87 

43-77 

45-52 

47-15 

48-67 

50-11 

52-75 

38 

30-72 

35-17 

38-71 

41-69 

44-66 

46-64 

48-77 

50-72 

52-54 

54-23 

55-83 

58-78 

40 

34-04 

38-97 

42-89 

46-20 

49-10 

51-69 

54-04 

56-20 

58-21 

60-09 

61-86 

65-12 

42 

37-53 

42-96 

47-29 

50-94 

54-13 

56-98 

59-58 

61-96 

64-18 

66-25 

68-21 

71-78 

44 

41-19 

47-15 

51-90 

55-91 

59-38 

62-54 

66-46 

68-00 

70-44 

72-71 

74-85 

78-79 

46 

45-02 

51-54 

56-72 

61-10 

64-88 

68-19 

71-43 

74-33 

76-69 

79-47 

81-81 

86-12 

48 

49-02 

56-11 

61-76 

66-53 

70-70 

74-42 

77-82 

80-94 

83-83 

86-53 

89-08 

93-78 

50 

53-19 

60-89 

67-02 

72-19 

76-71 

80-76 

84-44 

87-82 

9096 

93-89 

96-65 

101-7 

62 

57-55 

65-86 

72-48 

78-08 

83-00 

87-35 

90-25 

94-98 

98-40 

101-55 

104-5 

110-0 

54 

62-04 

71-02 

78-17 

84-20 

89-48 

94-20 

98-49 

102-4 

106-1 

109-5 

112-7 

118-7 

56 

66-72 

76-38 

84-07 

90-55 

96-23 

101-30 

105-9 

110-1 

114-1 

117-8 

121-2 

127-6 

58 

71-58 

81-93 

90-18 

97-14 

103-2 

108-6 

113-6 

118-2 

122-4 

126-3 

129-2 

136-7 

60 

76-60 

87-68 

96-50 

103-9 

110-4 

116-3 

121-6 

126-4 

131-0 

135-2 

1392 

146-6 

62 

81-79 

93-62 

103-04 

111-0 

117-96 

124-18 

129-81 

135-03 

139-86 

144-37 

148-6 

156-7 

64 

87-15 

99-84 

110-0 

118-3 

125-7 

132-3 

138-3 

143-9 

149-0 

153-82 

158-4 

166-7 

66 

92-68 

106-1 

116-8 

125-8 

133-6 

140-7 

147-3 

153-0 

158-5 

163-6 

168-4 

177-3 

68 

98-40 

112-6 

123-9 

133-6 

141-8 

149-4 

156-2 

162-4 

168-2 

173-6 

178-8 

188-2 

70 

104-26 

119-3 

131-3 

141-5 

150-4 

158-3 

165-5 

172-1 

178-2 

184-0 

189-4 

199-4 

72 

110-30 

126-2 

139-0 

149-7 

159-1 

167-4 

175-1 

182-1 

188-6 

194-7 

200-4 

211-0 

74 

116-5 

133-4 

146-8 

158-1 

167-9 

176-7 

185-4 

192-4 

199-2 

205-7 

211-6 

223-4 

76 

122-9 

140-7 

154-8 

166-8 

178-6 

186-6 

195-0 

202-9 

210-1 

246-9 

223-3 

235-1 

78 

129-4 

148-2 

163-1 

175-6 

186-7 

196-5 

205-4 

212-1 

221-4 

228-5 

235-2 

247-6 

80 

136-2 

155-8 

171-6 

184-8 

196-4 

206-7 

216-1 

224-8 

232-8 

240-4 

247-4 

260-5 

82 

143-0 

163-8 

180-2 

194-2 

206-2 

217-3 

226-9 

237-8 

244-6 

252-5 

260-0 

273-8 

84 

150-1 

171-8 

189-1 

203-8 

216-5 

227-9 

238-3 

247-8 

256-7 

265-0 

272-8 

287-1 

86 

157-4 

1SO-1 

198-2 

213-6 

227-0 

237-8 

247-4 

258-2 

269-1 

277-8 

286-0 

301-0 

88 

164-8    1188-6 

207-6 

223-6 

237-5 

250-2 

261-6 

272-0 

281-7 

290-8 

299-4 

315-2 

90 

172-3    1  197-3 

217-1 

233-9 

248-6 

261-7 

273-6 

284-5 

291-7 

304-2 

313-2 

329-7 

length  of  the  link  G  F,  aijd  divide  the  product  by  the  sum  of  the 
lengths  of  the  radius  bar  and  of  C  G.  The  quotient  is  the  length 
required. 

RULE  III.— (Figs.  3  and  4,  pages  246  and  247.)  To  find  the  length 
of  the  radius  bar  (Flf),  the  length  of  0  Gr  being  given. — Square  the 
length  of  C  G,  and  divide  it  by  the  length  of  D  G.  The  quotient 
is  the  length  required. 

RULE  IV.— (Figs.  3  and  4,  pages  246  and  247.)  To  find  the  length 
of  the  radius  bar,  the  horizontal  distance  of  its  centre  (If)  from  the 
main  centre  being  given. — To  this  given  horizontal  distance,  add  half 
the  versed  sine  (D  N)  of  the  arc  described  by  the  end  of  beam  (D). 
Square  this  sum.  Take  the  same  sum,  and  add  to  it  the  length  of 


244 


THE    PRACTICAL   MODEL   CALCULATOR. 


TABLE  of  Nominal  Horse  Power  of  High  Pressure  Engines. 


Diameter  of 
Cylinder  in 
Inches. 

LENGTH  OF  STROKE  IN  FEET. 

1 

u 

2 

2* 

3 

3£ 

4 

4£ 

5 

5* 

6 

7 

2 

•25 

•29 

•32 

•35 

•37 

•38 

•40 

•42 

•44 

•45 

•46 

•49 

2* 

•39 

•45 

•50 

•54 

•57 

•60 

•63 

•06 

•68 

•70 

•72 

•76 

3 

•57 

•65 

•72 

•78 

•83 

•87 

•91 

•95 

•98 

1-01 

1-04 

1-10 

3* 

•78 

•89 

•98 

1-06 

1-13 

1-19 

1-24 

1-29 

1-34 

1-38 

1-42 

1-49 

4 

1-02 

1-17 

1-29 

1-38 

1-47 

1-56 

1-62 

1-68 

1-74 

1-80 

1-86 

1-95 

4i 

1-29 

1-48 

1-63 

1-75 

1-86 

1-96 

2-05 

2-13 

2-21 

2-28 

2-35 

2-47 

5 

1-59 

1-83 

2-01 

2-16 

2-28 

2-43 

2-52 

2-64 

2-73 

2-82 

2-88 

3-06 

5* 

1-93 

2-21 

2-43 

2-62 

2-78 

2-93 

3-12 

3-18 

3-50 

3-42 

3-51 

3-69 

6 

2-28 

2-61 

2-88 

3-12 

3-30 

3-48 

3-66 

3-78 

3-93 

4-05 

4-17 

4-41 

6i 

2-69 

3-09 

3-39 

3-66 

3-90 

4-08 

4-23 

4-44 

4-62 

4-77 

4-89 

5-16 

7 

3-12 

3-57 

3-93 

4-23 

4-50 

4-74 

4-95 

5-16 

5-34 

5-52 

5-67 

5-97 

7* 

3-60 

4-11 

4-53 

4-86 

6-19 

5-46 

5-70 

5-94 

6-15 

6-33 

6-51 

6-87 

8 

4-08 

4'68 

5-16 

5-55 

5-88 

6-21 

6-48 

6-75 

6-99 

7-20 

7-41 

7-80 

8* 

4-62 

5-28 

5-82 

6-27 

6-63 

6-99 

7-32 

7-62 

7-89 

8-13 

8-37 

8-82 

9 

5-16 

5-91 

6-51 

7-02 

7-47 

7-86 

8-22 

8-52 

8-85 

9-12 

9-39 

9-90 

9* 

5-76 

6-60 

7-26 

7-80 

8-37 

8-76 

9-15 

9-51 

9-84 

10-17 

10-47 

10-01 

10 

6-39 

7-32 

•  8-04 

8-67 

9-21 

9-69 

10-14 

10-53 

10-92 

11-28 

11-61 

12-21 

10* 

7-05 

8-04 

8-88 

9-54 

10-14 

10-68 

11-16 

11-61 

12-03 

12-42 

12-78 

13-47 

11 

7-71 

8-85 

9-72 

10-47 

11-31 

11-73 

12-45 

12-75 

13-20 

13-62 

14-04 

14-76 

11* 

8-43 

9-66 

10-62 

11-46 

12-15 

12-78 

13-80 

13-92 

14-61 

14-91 

15-33 

16-14 

12 

9-18 

10-53 

11-58 

12-41 

13-26 

13-95 

14-58 

15-18 

15-72 

16-23 

16-71 

17-58 

12* 

9-96 

11-40 

12-57 

13-53 

14-37 

15-15 

15-84 

16-47 

17-04 

17-58 

18-12 

19-08 

13 

10-80 

12-36 

13-59 

14-64 

15-57 

16-38 

16-92 

17-82 

18-45 

19-05 

19-59 

21-64 

13i 

11-64 

13-32 

14-64 

15-78 

16-77 

17-67 

18-48 

19-20 

19-89 

20-52 

21-15 

22-26 

14 

12-51 

14-31 

15-75 

16-98 

18-03 

18-99 

19-86 

20-64 

21-39 

22-08 

22-74 

23-94 

14* 

13-41 

15-36 

16-92 

18-21 

19-35 

20-37 

21-30 

22-14 

22-95 

23-70 

24-39 

25-62 

15 

14-31 

16-44 

18-09 

19-50 

20-70 

21-81 

22-80 

23-70 

24-57 

25-35 

26-10 

27-48 

16 

16-35 

18-69 

20-58 

22-17 

23-58 

24-81 

25-95 

26-97 

27-93 

28-83 

29-70 

31-26 

17 

18-45 

21-12 

23-25 

25-05 

26-58 

28-02 

29-28 

30-45 

31-56 

32-55 

33-57 

35-28 

18 

20-67 

23-67 

26-04 

28-08 

29-82 

31-41 

32-82 

34-14 

35-37 

36-51 

37-59 

39-57 

19 

23-04 

26-37 

29-04 

31-26 

33-51 

34-98 

36-57 

38-04 

39-39 

40-68 

41-88 

44-07 

20 

25-53 

29-22 

32-16 

34-65 

36-81 

38-76 

40-53 

42-15 

43-65 

45-06 

46-38 

48-84 

22 

30-90 

35-37 

38-91 

41-94 

44-55 

46-89 

49-86 

51-90 

52-95 

54-54 

56-13 

59-10 

24 

36-78 

42-09 

46-32 

49-89 

53-01 

55-83 

58-35 

60-69 

62-85 

64-89 

66-81 

70-32 

26 

43-17 

49-38 

54-36 

58-56 

62-25 

65-52 

67-68 

71-25 

73-80 

76-17 

78-42 

82-53 

28 

•50-04 

57-27 

63-06 

67-92 

72-18 

75-99 

79-44 

82-62 

85-56 

88-32 

90-93 

95-70 

30 

57-45 

65-76 

72-39 

77-97 

82-86 

87-21 

91-20 

94-83 

98-22 

101-40 

104-4 

109-9 

32 

65-37 

74-88 

82-53 

88-71 

94-26 

99-24 

103-7 

107-9 

111-8 

115-4 

118-7 

125-0 

34 

73-80 

84-48 

92-9 

100-22 

106-3 

112-0 

117-1 

121-8 

126-2 

130-2 

134-0 

141-1 

36 

82-71 

94-68 

104-2 

112-2 

119-3 

125-6 

131-3 

136-5 

141-4 

146-0 

150-3 

158-2 

38 

92-16 

105-5 

116-1 

125-0 

134-0 

136-9 

146-3 

152-1 

157-6 

162-7 

167-5 

176-3 

40 

102-1 

116-9 

129-6 

128-6 

147-3 

155-1 

162-1 

168-6 

174-6 

180-2 

185-6 

195-3 

42 

112-6 

128-9 

141-8 

152-8 

162-4 

170-9 

178-7 

185-9 

192-5 

198-7 

204-6 

215-3 

44 

123-5 

141-4 

155-7 

167-7 

178-1 

187-6 

199-4 

204-0 

211-3 

218-1 

224-5 

236-3 

46 

135-0 

154-6 

H70-1 

183-3 

194-6 

204-6 

214-3 

223-0 

230-0 

238-4 

245-4 

258-3 

48 

147-0 

168-3 

185-3 

199-6 

212-1 

223-2 

233-4 

242-8 

251-5 

259-6 

267-2 

281-3 

50 

159-6 

182-6 

201-0 

216-5 

230-1 

242-3 

253-3 

263-4 

272-9 

281-6 

289-9 

305-1 

52 

172-6 

197-6 

217-4 

234-2 

249-0 

262-0 

270-7 

284-9 

295-2 

304-6 

313-5 

330-0 

54 

186-1 

213-0 

234-5 

252-6 

268-4 

282-6 

295-4 

307-2 

318-3 

328-5 

338-1 

356-1 

56 

200-1 

229-1 

252-2 

271-6 

288-7 

303-9 

317-7 

330.3 

342-3 

353-4 

363-6 

382-8 

58 

214-7 

245-8 

270-5 

291-4 

309-6 

325-8 

340-8 

354-6 

367-2 

378-9 

389-7 

410-1 

60 

229-8 

263-0 

289-5 

311-7 

331-2 

348-9 

364-8 

379-2 

393-0 

405-6 

417-6 

439-5 

the  beam  (C  D).  Divide  the  square  previously  found  by  this  last 
sum,  and  the  quotient  is  the  length  sought. 

RULE  N.—(Figs.  5  and  6,  pages  247,  248.)— To  find  the  length 
of  the  radius  bar,  0  G-  and  P  Q  being  given. — Square  C  G,  and 
multiply  the  square  by  the  length  of  the  side  rod  (P  D) :  call  this 
product  A.  Multiply  Q  D  by  the  length  of  the  side  lever  (C  D). 
I^rom  this  product  subtract  the  product  of  D  P  into  C  G,  and  divide 
A  by  the  remainder.  The  quotient  is  the  length  required. 

RULE  VI.— (Figs.  5  and  6,  pages  24T,  248.)  To  find  the  length  of 
the  radius  bar  ;  P  $,  and  the  horizontal  distance  of  the  centre  H  of 
the  radius  bar  from  the  main  centre  being  given. — To  the  given  hori- 
zontal distance  add  half  the  versed  sine  (D  N)  of  the  arc  described 


THE   STEAM   ENGINE. 
Fig.  1. 


245 


ii  C 


Fig.  2. 


by  the  extremity  (D)  of  the  side  lever.  Square  this  sum  and  mul- 
tiply the  square  by  the  length  of  the  side  rod  (P  D).  Call  this  pro- 
duct A.  Take  the  same  horizontal  distance  as  before  added  to  the 
same  half  versed  sine  (D  N),  and  multiply  the  sum  by  the  length  of 
the  side  rod  (P  D) :  to  the  product  add  the  product  of  the  length  of 
v2 


246 


THE  PRACTICAL  MODEL  CALCULATOR. 
Fig.  3. 


the  side  lever  C  D  into  the  length  of  Q  D,  and  divide  A  by  the 
sum.  The  quotient  will  be  the  length  required. 

When  the  centre  H  of  the  radius  has  its  position  determined, 
rules  4  and  6  will  always  give  the  length  of  the  radius  bar  F  H. 
To  get  the  length  of  C  G,  it  will  only  be  necessary  to  draw  through 
the  point  F  a  line  parallel  to  the  side  rod  D  P,  and  the  point  where 
that  line  cuts  D  C  will  be  the  position  of  the  pin  G. 

In  using  these  formulas  and  rules,  the  dimensions  must  all  be 
taken  in  the  same  measure ;  that  is,  either  all  in  feet,  or  all  in 
inches  ;  and  when  great  accuracy  is  required,  the  corrections  given 
in  Table  (A)  must  be  added  to  or  subtracted  from  the  calculated 
length  of  the  radius  bar,  according  as  it  is  less  or  greater  than  the 
length  of  C  G,  the  part  of  the  beam  that  works  it. 

1.  RULE  4. — Let  the  horizontal  distance  (M  C)  of  the  centre  (H) 


THE   STEAM    ENGINE. 
Fig.  4. 


247 


Fig.  6. 


of  the  radius  bar  from  the  main  centre  be  equal  to  51  inches ;  the 
half  versed  sine  D  N  =  3  inches,  and  D  C  =  126  inches ;  then  by 
the  rule  we  will  have 


(51  4-  3)2  (54)2       2916 

51  +  8  +  126  =  :  180"  =  '  W  =  hes> 

which  is  the  required  length  of  the  radius  bar  (F  H). 


248 


THE  PRACTICAL  MODEL  CALCULATOR. 
Fig.  6. 


2.  RULE  5. — The  following  dimensions  are  those  of  the  Red  Rover 
steamer :    C  G  =  32  D  P  =  94  Q  D  =  74  C  D  =  65  P  Q  =  20. 

By  the  rule  we  have,  A  =  (32)2  x  94  =  96256  and 

96256 96256  _ 

74  x  65  -  94  x  32  ~     1802  :~  534> 
which  is  the  required  length  of  the  radius  bar. 

3.  RULE  6. — Take  the  same  data  as  in  the  last  example,  only 
supposing  that  C  G  is  not  given,  and  that  the  centre  H  is  fixed  at 
a  horizontal  distance  from  the  main  centre,  equal  to  83-5  inches. 
Then  the  half  versed  sine  of  the  arc  D'  D  D"  will  be   about  2 
inches,  and  we  will  have  by  the  rule 

A  =  (83-5  +  2)2  x  94  =  705963-5  and 


705963-5 


85-5  x  94  +  65  x  74         1284-7 
the  required  length  of  the  radius  bar  in  this  case. 

TABLE  (A). 


54-8  inches, 


TT1   TT                        ' 

This  column  gives  -~-^  when 
(j  (y 

CG 

C  G  is  the  greater,  and  -^-rr 

when  F  H  is  the  greater. 

Correction  to  be  added  to  or 
subtracted  from  the  calcu- 
lated  length  of  the  radius 
bar,  in  decimal  parts  of  its 
calculated  length. 

'f 

1-0 
•9 
•8 
•1 
•6 
•5 
•4 

0 
•0034 
•0075 
•0163 
•0270 
•0452 
•0817 

THE   STEAM   ENGINE.  249 

C  G 

In  both  of  the  last  two  examples  jj^  =  -6  nearly.  The  correc- 
tion found  by  Table  (A),  therefore,  would  be  54  X  -027  =  1-458 
inches,  which  must  be  subtracted  from  the  lengths  already  found 
for  the  radius  bar,  because  it  is  longer  than  C  G.  The  corrected 
lengths  will  therefore  be 

In  example  2 F  H  =  51*94  inches. 

In  example  3 F  H  =  53-34  inches. 

RULE. —  To  find  the  depth  of  the  main  beam  at  the  centre. — Divide 
the  length  in  inches  from  the  centre  of  motion  to  the  point  where 
the  piston  rod  is  attached,  by  the  diameter  of  the  cylinder  in 
inches  ;  multiply  the  quotient  .by  the  maximum  pressure  in  pounds 
per  square  inch  of  the  steam  in  the  boiler ;  divide  the  product  by 
202  for  cast  iron,  and  236  for  malleable  iron :  in  either  case,  the 
cube  root  of  the  quotient  multiplied  by  the  diameter  of  the  cylinder 
in  inches  gives  the  depth  in  inches  of  the  beam  at  the  centre  of 
motion.  To  find  the  breadth  at  the  centre. — Divide  the  depth  in 
inches  by  16 ;  the  quotient  is  the  breadth  in  inches. 

An  engine  beam  is  three  times  the  diameter  of  the  cylinder,  from 
the  centre  to  the  point  where  the  piston  rod  acts  on  it ;  the  force 
of  the  steam  in  the  boiler  when  about  to  force  open  the  safety 
valve  is  10  Ibs.  per  square  inch.  Required  the  depth  and  breadth 
when  the  beam  is  of  cast  iron.  • 

In  this  case  n  =  3,  and  P  =  10,  and  therefore 

I«*     $*H£ 

The  breadth  =  —  D  =  -03  D. 
16 

It  will  be  observed  that  our  rule  gives  the  least  value  to  the 
depth.  In  actual  practice,  however,  it  is  necessary  to  make  allow- 
ance for  accidents,  or  for  faultiriess  in  the  materials.  This  may  be 
done  by  making  the  depth  greater  than  that  determined  by  the 
rule  ;  or,  perhaps  more  properly,  by  taking  the  pressure  of  the  steam 
much  greater  than  it  can  ever  possibly  be.  As  for  the  dimensions 
of  the  other  parts  of  the  beam,  it  is  obvious  that  they  ought  to 
diminish  towards  the  extremities ;  for  the  power  of  a  beam  to  resist 
a  cross  strain  varies  inversely  as  its  length.  The  dimensions  may 
be  determined  from  the  formula/ b  d2  =  6  W  I. 

To  apply  the  formula  to  cranks,  we  may  assume  the  depth  at  the 
shaft  to  be  equal  to  n  times  the  diameter  of  the  shaft ;  hence,  if 
m  X  D  be  the  diameter  of  the  shaft,  the  depth  of  the  crank  will 
be  n  X  m  x  D.  Substituting  this  in  the  formula  fb  d2  —  6  W  ?, 
and  it  becomes  fb  X  n2  X  m2  X  D2  =  6  W  I.  Now,  as  before, 
W  =  -7854  X  P  x  D2,  so  that  the  formula  becomes  /  x  b  X  riz  x 
m2  =  4-7124  X  P  x  I.  The  value  of  n  is  arbitrary.  In  practice 
it  may  be  made  equal  to  1J  or  1-5.  Taking  this  value,  then,  for 


250  THE  PRACTICAL  MODEL  CALCULATOR. 

cast  iron,  the  formula  becomes  15300  x  b  x  f  X  w2  =  4-7124  x 
P  X  I,  or  7305  m2  b  =  P  I ;  but  if  L  denote  the  length  of  the  crank 
in  feet,  the  formula  becomes  609  mz  b  =  PL,  and  .*.  b  =  P  x 
L  -T-  609  m2.  This  formula  may  be  put  into  the  form  of  a  rule, 
thus : — 

RULE. — To  find  the  breadth  at  the  shaft  when  the  depth  is  equal 
to  1J  times  the  diameter  of  the  shaft. — Divide  the  square  of  the 
diameter  of  the  shaft  in  inches  by  the  square  of  the  diameter  of 
the  cylinder ;  multiply  the  quotient  by  609,  and  reserve  the  pro- 
duct for  a  divisor ;  multiply  the  greatest  elastic  force  of  the  steam 
in  Ibs.  per  square  inch  by  the  length  of  the  crank  in  feet,  and 
divide  the  product  by  the  reserved  divisor:  the  quotient  is  the 
breadth  of  the  crank  at  the  shaft.  - 

A  crank  shaft  is  J  the  diameter  of  the  cylinder;  the  greatest 
possible  force  of  the  steam  in  the  boiler  is  20  Ibs.  per  square  inch ; 
and  the  length  of  the  shaft  is  3  feet.  Required  the  breadth  of  the 
crank  at  the  shaft  when  its  depth  is  equal  to  1J  times  the  diameter 
of  the  shaft. 

In  this  case  m  =  J-,  so  that  the  reserved  divisor  —  y^-  =  38 : 
again,  elastic  force  of  steam  in  Ibs.  per  square  inch  =  20  Ibs. ; 

O     v^     OA 

hence  width  of  crank  =  — QK —  =  1-6  inches  nearly. 

RULE. — To  find  the  diameter  of  a  revolving  shaft. — Form  a 
reserved  divisor  thus  :  multiply  the  number  of  revolutions  which 
the  shaft  makes  for  each  double  stroke  of  the  piston  by  the  number 
1222  for  cast  iron,  and  the  number  1376  for  malleable  iron.  Then 
divide  the  radius  of  the  crank,  or  the  radius  of  the  wheel,  by  the 
diameter  of  the  cylinder  ;  multiply  the  quotient  by  the  greatest 
pressure  of  the  steam  in  the  boiler  expressed  in  Ibs.  per  square 
inch ;  divide  the  product  by  the  reserved  divisor  ;  extract  the  cube 
root  of  the  quotient,  and  multiply  the  result  by  the  diameter  of 
the  cylinder  in  inches.  The  product  is  the  diameter  of  the  shaft 
in  inches. 

STRENGTH    OP   RODS    WHEN   THE    STRAIN    IS  WHOLLY   TENSILE  J    SUCH   AS 
THE   PISTON   ROD   OF    SINGLE   ACTING   ENGINES,    PUMP   RODS,  ETC. 

RULE. — To  find  the  diameter  of  a  rod  exposed  to  a  tensile  force 
only. — Multiply  the  diameter  of  the  piston  in  inches  by  the  square 
root  of  the  greatest  elastic  force  of  the  steam  in  the  boiler  esti-* 
mated  in  Ibs.  per  square  inch ;  the  product,  divided  by  95,  is  the 
diameter  of  the  rod  in  inches. 

Required  the  diameter  of  the  transverse  section  of  a  piston  rod 
in  a  single  acting  engine,  when  the  diameter  of  the  cylinder  is  50 
inches,  and  the  greatest  possible  force  of  the  steam  in  the  boiler  is 
16  Ibs.  per  square  inch.  Here,  according  to  the  formula, 

50     200 

d  =        s/  16  =  --  =  2-1  inches. 


THE    STEAM    ENGINE.  251 

RULE. — To  find  the  strength  of  rods  alternately  extended  and 
compressed,  such  as  the  piston  rods  of  double  acting  engines. — Mul- 
tiply the  diameter  of  the  piston  in  inches  by  the  square  root  of  the 
maximum  pressure  of  the  steam  in  Ibs.  per  square  inch  ;  divide  the 
product  by 

47  for  cast  iron, 

50  for  malleable  iron. 

This  rule  applies  to  the  piston  rods  of  double  acting  engines, 
parallel  motion  rods,  air-pump  and  force-pump  rods,  and  the  like. 
The  rule  may  also  be  applied  to  determine  the  strength  of  connect- 
ing rods,  by  taking,  instead  of  P,  a  number  Pr,  such  that  Pr  X  sine 
of  the  greatest  angle  which  the  connecting  rod  makes  with  the 
direction  =  P. 

Supposing  the  greatest  force  of  the  steam  in  the  boiler  to  be  16 
Ibs.  per  square  inch,  and  the  diameter  of  the  cylinder  50  inches ; 
required  the  diameter  of  the  piston  rod,  supposing  the  engine  to  be 
double  acting.  In  this  case 

for  cast  iron  d  =  —  \/  P  = =  5  inches  nearly ; 

j)     

for  malleable  iron  d  =  —  v/  P  =  4  inches. 
50 

The  pressure,  however,  is  always  taken  in  practice  at  more  than  16 
Ibs.  If  the  pressure  be  taken  at  25  Ibs.,  the  diameter  of  a  malle- 
able iron  piston  rod  will  be  5  inches,  which  is  the  usual  proportion. 
Piston  rods  are  never  made  of  cast  iron,  but  air-pump  rods  are 
sometimes  made  of  brass,  and  the  connecting  rods  of  land  engines 
are  cast  iron  in  most  cases. 

FORMULAS  FOR  THE   STRENGTH  OF  VARIOUS   PARTS  OF  MARINE  ENGINES. 

The  following  general  rules  give  the  dimensions  proper  for  the 
parts  of  marine  engines,  and  we  shall  recapitulate,  with  all  possible 
brevity,  the  data  upon  which  the  denominations  rest. 

Let  pressure  of  the  steam  in  boiler  =  p  Ibs.  per  square  inch, 
Diameter  of  cylinder  =  D  inches, 
Length  of  stroke         =  2  R  inches. 

The  vacuum  below  the  piston  is  never  complete,  so  that  there 
always  remains  a  vapour  of  steam  possessing  a  certain  elasticity. 
We  may  suppose  this  vapour  to  be  able  to  balance  the  weight  of  the 
piston.  Hence  the  entire  pressure  on  the  square  inch  of  piston  in 
Ibs.  =  p  +  pressure  of  atmosphere  =  15  -f  p.  We  shall  substi- 
tute P  for  15  +  p.  Hence 

Entire  pressure  on  piston  in  Ibs.  =  -7854  X  (15  -f  p)  X  D2 

=  -7854  x  P  x  D2. 

The  dimensions  of  the  paddle-shaft  journal  may  be  found  from 
the  following  formulas,  which  are  calculated  so  that  the  strain  in 
ordinary  working  =  f  elastic  force. 

Diameter  of  paddle-shaft  journal  =  -08264  {R  x  P  X  D2}^ 
Length  of  ditto  =  1 J  X  diameter. 


252          THE  PRACTICAL  MODEL  CALCULATOR. 

The  dimensions  of  the  several  parts  of  the  crank  may  be  found 
from  the  following  formulas,  which  are  calculated  so  that  the  strain 
in  ordinary  working  =  one-half  the  elastic  force  ;  and  when  one 
paddle  is  suddenly  brought  up,  the  strain  at  shaft  end  of  crank  =  f 
elastic  force,  the  strain  at  pin  end  of  crank  =  elastic  force. 

Exterior  diameter  of  large  eye  =  diameter  of  paddle-shaft  -j- 
|  D  [P  x  1-561  x  R2  +  '00494  x  D2  x  P2]p 

1"  75-59  x  S&~ 

Length  of  ditto  =  diameter  of  paddle  shaft. 
Exterior  diameter  of  small  eye  =  diameter  of  crank  pin 
•02521  x  v'T  x  D. 

Length  of  ditto  =  -0375  X  v'T  X  D. 
Thickness  of  web  at  paddle  centre  = 


D2  x  P  x  V  {1-561  x  R2  +  -00494  x  D2  x  P 

~~  .  9000 

Breadth  of  ditto  =  2  X  thickness. 
Thickness  of  web  at  pin  centre  —  -022  X  vHP  X  D. 
Breadth  of  ditto  =  f  X  thickness. 

As  these  formulas  are  rather  complicated,  we  may  show  what 
they  become  when  p  —  10  or  P  =  25. 

Exterior  diameter  of  large  eye  =  diameter  of  paddle  shaft  + 


D  V  (1-561  x  R2  -f  -1235  x  D2) 

15-12  x  v/R 

Length  of  ditto  =  diameter  of  paddle  shaft. 

Exterior  diameter  of  small  eye  =  equal  diameter  of  crank  pin  + 
•126  x  D. 

Length  of  ditto  =  -1875  X  D. 

Thickness  of  web  at  pin  centre  =  *11  X  D. 

Breadth  of  ditto  =  f  X  thickness  of  web. 

The  dimensions  of  the  crank  pin  journal  may  be  found  from  the 
following  formulas,  which  are  calculated  so  that  strain  when  bear- 
ing at  outer  end  =  elastic  force,  and  in  ordinary  working  strain  = 
one-third  of  elastic  force. 

Diameter  of  crank-pin  journal  =  -02836  X  v'T  x  D. 

Length  of  ditto  =  f  X  diameter. 

The  dimensions  of  the  several  parts  of  the  cross  head  may  be  found 
from  the  following  formulas,  in  which  we  have  assumed,  for  the 
purpose  of  calculation,  the  length  =  1*4  X  D.  -The  formulas 

have  been  calculated  so  as  to  give  the  strain  of  web  =  ^-^^  x 

' 


elastic  force  ;  strain  of  journal  in  ordinary  working  =  __  x  elastic 

2*00 

force,  and  when  bearing  at  outer  end  =  ^-=r^  x  elastic  force. 

' 


THE    STEAM    ENGINE.  253 

Exterior  diameter  of  eye  =  diameter  of  hole  +  -02827  X  P^  X  D. 

Depth  of  ditto  =  -0979  x  P*  x  IX_ 
Diameter  of  journal  =  -01716  X  V  P  X  D. 
Length  of  ditto  =  f  diameter  of  journal. 

Thickness  of  web  at  middle  =  -0245  X  P^  x  D. 
Breadth  of  ditto  =  -09178  x  P*  x  D. 
Thickness  of  web  at  journal  =  -0122  X  P2  X  D. 

Breadth  of  ditto  =  -0203  X  P*  x  D. 

The  dimensions  of  the  several  parts  of  the  piston  rod  may  be 
found  from  the  following  formulas,  which  are  calculated  so  that  the 
strain  of  piston  rod  =  {  elastic  force. 

7~P  vv  ry 

Diameter  of  the  piston  rod  =  — — — — 

50 

Length  of  part  in  piston  =  -04  X  D  X  P. 

Major  diameter  of  part  in  crosshead  =  -019  X  v/P  X  D. 

Minor  diameter  of  ditto  =  -018  X  v/P  X  D. 

Major  diameter  of  part  in  piston  =  -028  X  v/P  X  D. 

Minor  diameter  of  ditto  =  -023  X  v/P  X  D. 

Depth  of  gibs  and  cutter  through  crosshead  =  -0358  X  P^  X  D. 

Thickness  of  ditto  =  -007  X  P^  X  D. 

Depth  of  cutter  through  piston  =  -017  X  v/P  X  D. 

Thickness  of  ditto  =  -007  X  P^  X  D. 

The  dimensions  of  the  several  parts  of  the  connecting  rod  may 
be  found  from  the  following  formulas,  which  are  calculated  so  that 
the  strain  of  the  connecting  rod  and  the  strain  of  the  strap  are  both 
equal  to  one-sixth  of  the  elastic  force.  x 

Diameter  of  connecting  rod  at  ends  =  -019  X  P2  X  D. 

Diameter  of  ditto  at  middle  =  {1  -f  -0035  X  length  in  inches} 
x  -019  x  v/P  x  D. 

Major  diameter  of  part  in  crosstail  =  -0196  X  P*  X  D. 
Minor  ditto  =  -018  X  P2  X  D. 
Breadth  of  butt  =  -0313  X  P*  x  D. 

Thickness  of  ditto  =  -025  x  P^  X  D. 

Mean  thickness  of  strap  at  cutter  =  -00854  X  v/P  X  D. 

Ditto  above  cutter  =  -00634  X  v/P  X  D. 

Distance  of  cutter  from  end  of  strap  =  -0097  X  v/P  X  D. 

Breadth  of  gibs  and  cutter  through  crosstail  =  -0358  X  P^  x  D. 
Breadth  of  gibs  and  cutter  through  butt  =  -022  x  P2  X  D. 

Thickness  of  ditto  =  -00564  X  P2  X  D. 
w 


254  THE    PRACTICAL    MODEL   CALCULATOR. 

The  dimensions  of  the  several  parts  of  the  side  rods  may  be 
found  from  the  following  formulas,  which  are  calculated  so  as  to 
make  the  strain  of  the  side  rod  =  one-sixth  of  elastic  force,  and 
the  strains  of  strap  and  cutter  =  one-fifth  of  elastic  force. 

Diameter  of  cylinder  side  rods  at  ends  =  -0129  x  P^  x  D. 
Diameter  of  ditto  at  middle  =  (1  -f  -0035  X  length  in  inches). 

x  -0129  x  pi  x  D. 
Breadth  of  butt  =  -0154  x  P^  x  D. 
Thickness  of  ditto  =  -0122  X  P*  x  D. 

Diameter  of  journal  at  top  end  of  side  rod  =  -01716  X  P¥  x  D. 
Length  of  journal  at  top  end  =  f  diameter. 

Diameter  of  journal  at  bottom  end  =  '014  X  P¥  X  D. 

Length  of  ditto  =  -0152  x  P^  X  D. 

Mean  thickness  of  strap  at  cutter  =  -00643  X  P*  x  D. 

Ditto  below  cutter  =  -0047  X  P*  x  D. 

Breadth  of  gibs  and  cutter  =  -016  X  P*  x  D. 

Thickness  of  ditto  =  -0033  x  P*  x  D. 

The  dimensions  of  the  main  centre  journal  may  be  found  from 
the  following  formulas,  which  are  calculated  so  as  to  make  the 
strain  in  ordinary  working  =  one  half  elastic  force. 

Diameter  of  main  centre  journal  =  -0367  X  P2  X  D. 

Length  of  ditto  =  f  X  diameter. 

The  dimensions  of  the  several  parts  of  the  air-pump  may  be 
found  from  the  corresponding  formulas  given  above,  by  taking  for 
D  another  number  d  the  diameter  of  air-pump. 

DIMENSIONS    OF   THE    SEVERAL   PARTS    OF    FURNACES    AND   BOILERS. 

Perhaps  in  none  of  the  parts  of  a  steam  engine  does  the  practice 
of  engineers  vary  more  than  in  those  connected  with  furnaces  and 
boilers.  There  are,  no  doubt,  certain  proportions  for  these,  as  well 
as  for  the  others,  which  produce  the  maximum  amount  of  useful 
effect  for  particular  given  purposes  ;  but  the  determination  of  these 
proportions,  from  theoretical  considerations,  has  hitherto  been  at- 
tended with  insuperable  difficulties,  arising  principally  from  our  im- 
perfect knowledge  of  the  laws  of  combustion  of  fuel,  and  of  the  laws 
according  to  which  caloric  is  imparted  to  the  water  in  the  boiler. 
In  giving,  therefore,  the  following  proportions  for  the  different 
parts,  we  desire  to  have  it  understood  that  we  do  not  affirm  them 
to  be  the  best,  absolutely  considered ;  we  give  them  only  as  the 
average  practice  of  the  best  modern  constructors.  In  most  of  the 
cases  we  have  given  the  average  value  per  nominal  horse  power. 
It  is  well  known  that  the  tern}  horse  power  is  a  conventional  unit 
for  measuring  the  size  of  steam  engines,  just  as  a  foot  or  a  mile  is 


THE   STEAM   ENGINE.  255 

a  unit  for  the  measurement  of  extension.  There  is  this  difference, 
however,  in  the  two  cases,  that  whereas  the  length  of  a  foot  is 
fixed  definitively,  and  is  known  to  every  one,  the  dimensions  proper 
to  an  engine  horse  power  differ  in  the  practice  of  every  different 
maker :  and  the  same  kind  of  confusion  is  thereby  introduced  into 
engineering  as  if  one  person  were  to  make  his  foot-rule  eleven 
inches  long,  and  another  thirteen  inches.  It  signifies  very  little 
what  a  horse  power  is  defined  to  be ;  but  when  once  defined,  the 
measurement  should  be  kept  inviolable.  The  question  now  arises, 
what  standard  ought  to  be  the  accepted  one.  For  our  present  pur- 
pose, it  is  necessary  to  connect  by  a  formula  the  three  quantities, 
nominal  horses  power,  length  of  stroke,  and  diameter  of  cylinder. 
With  this  intention, 

Let  S  =  length  of  stroke  in  feet, 

d  =  diameter  of  cylinder  in  inches  ; 

Then  the  nominal  horse  power  = -r* —  nearly. 

I.  Area  of  Fire  Grate. — The  average  practice  is  to  give  '55 
square  feet  for  each  nominal  horse  power.  Hence  the  following 
rule : 

RULE  1. — To  find  the  area  of  the  fire  grate. — Multiply  the  num- 
ber of  horses  power  by  -55  ;  the  product  is  the  area  of  the  fire  grate 
in  square  feet. 

Required  the  total  area  of  the  fire  grate  for  an  engine  of  400 
horse  power.  Here  total  area  of  fire  grate  in  square  feet  =  400  X 
•55  =  220. 

A  rule  may  also  be  found  for  expressing  the  area  of  the  fire  grate 
in  terms  of  the  length  of  stroke  and  the  diameter  of  the  cylinder. 
For  this  purpose  we  have, 

•55  X  d2  X  -^S               d2  X  &S  „ 
total  area  of  fire  grate  =  -       — **. feet  =  ?w> feet. 

This  formula  expressed  in  words  gives  the  following  rule. 

RULE  2. —  To  find  the  area  of  fire  grate. — Multiply  the  cube  root 
of  the  length  of  stroke  in  feet  by  the  square  of  the  diameter  in  in- 
ches ;  divide  the  product  by  86 ;  the  quotient  is  the  area  of  fire 
grate  in  square  feet. 

Required  the  total  area  of  the  fire  grate  for  an  engine  whose 
stroke  =  8  feet,  and  diameter  of  cylinder  =  50  inches. 

Here,  according  to  the  rule, 

502  x  ^8       2500  x  2 
total  area  of  fire  grate  in  square  feet  = -^ = TTTJ = 

5000 

-gg-  =  59  nearly. 

In  order  to  work  this  example  by  the  first  rule,  we  find  the 
nominal  horse  power  of  the  engine  whose  dimensions  we  have  spe- 
cified is  104-3 ;  hence, 

total  area  of  fire  grate  in  square  feet  =  106-4  X  -55  =  58*5. 


256  THE    PRACTICAL    MODEL    CALCULATOR. 

With  regard  to  these  rules  we  may  remark,  not  only  that  they 
are  founded  on  practice,  and  therefore  empyrical,  but  they  are  only 
applicable  to  large  engines.  When  an  engine  is  very  small,  it  re- 
quires a  much  larger  area  of  fire  grate  in  proportion  to  its  size  than 
a  larger  one.  This  depends  upon  the  necessity  of  having  a  certain 
amount  of  fire  grate  for  the  proper  combustion  of  the  coal. 

II.  Length  of  Furnace.  —  The  length  of  the  furnace  differs  con- 
siderably, even  in  the  practice  of  the  same  engineer.     Indeed,  all 
the  dimensions  of  the  furnace  depend  to  a  certain  extent  upon  the 
peculiarity  of  its  position.     From  the  difficulty  of  firing  long  fur- 
naces efficiently,  it  has  been  found  more  beneficial  to  restrict  the 
length  of  the  furnace  to  about  six  feet  than  to  employ  furnaces  of 
greater  length. 

III.  Height  of  Furnace  above  Bars.  —  This  dimension  is  variable, 
but  it  is  a  common  practice  to  make  the  height  about  two  feet. 

IV.  Capacity  of  Furnace   Chamber  above  Bars.  —  The  average 
per  horse  power  may  be  taken  at  1-17  feet.     Hence  the  following 
rule  : 

RULE.  —  To  find  the  capacity  of  furnace  chamber  above  bars.  — 
Multiply  the  number  of  nominal  horses  power  by  1-17  ;  the  pro- 
duct is  the  capacity  of  furnace  chambers  above  bars  in  cubic  feet. 

V.  Areas  of  Flues  or  Tubes  in  smallest  part.  —  The  average  value 
of  the  area  per  horse  power  is  11*2  sq.  in.     Hence  we  have  the  fol- 
lowing rule  : 

RULE.  —  To  find  the  total  area  of  the  flues  or  tubes  in  smallest 
part.  —  Multiply  the  number  of  horse  power  by  11-2  ;  the  product 
is  the  total  area  in  square  inches  of  flues  or  tubes  in  smallest  part. 

Required  total  area  of  flues  or  tubes  for  the  boiler  of  a  steam  en- 
gine when  the  horse  power  =  400. 

For  this  example  we  have,  according  to  the  rule, 

Total  area  in  square  inches  =  400  X  11  '2  =  4480. 

We  may  also  find  a  very  convenient  rule  expressed  in  terms  of 
the  stroke  and  the  diameter  of  cylinder.  Thus, 

11-2  x  d2  x 
Total  area  of  tubes  or  flues  in  square  inches  =  -       —  j=  — 


4       ' 

VI.  Effective  Heating  Surface.  —  The  effective  heating  surface  of 
flue  boilers  is  the  whole  of  furnace  surface  above  bars,  the  whole 
of  tops  of  flues,  half  the  sides  of  flues,  and  none  of  the  bottoms  ; 
hence  the  effective  flue  surface  is  about  half  the  total  flue  surface. 
In  tubular  boilers,  however,  the  whole  of  the  tube  surface  is  reckoned 
effective  surface. 

EFFECTIVE  HEATING  SURFACE  OF  FLUE  BOILERS. 

RULE  1.  —  To  find  the  effective  heating  surface  of  marine  flue 
boilers  of  large  size.  —  Multiply  the  number  of  nominal  horse 
power  by  5  ;  the  product  is  the  area  of  effective  heating  surface  in 
square  feet. 


THE    STEAM    ENGINE.  257 

Required  the  effective  heating  surface  of  an  engine  of  400  nomi- 
nal horse  power. 

In  this  case,  according  to  the  rule,  effective  heating  surface  in 
square  feet  =  400  X  5  =  2000. 

The  effective  heating  surface  may  be  expressed  in  terms  of  the 
length  of  stroke  and  the  diameter  of  the  cylinder. 

RULE  2. —  To  find  the  total  effective  heating  surface  of  marine 
flue  boilers. — Multiply  the  square  of  the  diameter  of  cylinder  in 
inches  by  the  cube  root  of  the  length  of  stroke  in  feet ;  divide  the 
product  by  10 :  the  quotient  expresses  the  number  of  square  feet 
of  effective  heating  surface. 

Required  the  amount  of  effective  heating  surface  for  an  engine 
whose  stroke  =  8  ft.,  and  diameter  of  cylinder  =  50  inches. 

Here,  according  to  Rule  2,  effective  heating  surface  in  square  feet 

502  x  ^8       2500  x  2       5000 

"To~         "io"       ~io~ 

To  solve  this  example  according  to  the  first  rule,  we  have  the 
nominal  horse  power  of  the  engine  equal  to  106-4.  Hence,  ac- 
cording to  Rule  2,  total  effective  heating  surface  in  square  feet  = 
106-4  x  4-92  =  523J. 

EFFECTIVE   HEATING   SURFACE    OF   TUBULAR   BOILERS. 

vThe  effective  heating  surface  of  tubular  boilers  is  about  equal  to 
the  total  heating  surface  of  flue  boilers,  or  is  double  the  effective 
surface  ;  but  then  the  total  tube  surface  is  reckoned  effective  sur- 
face. 

It  appears  that  the  total  heating  surface  of  flue  and  tubular  ma- 
rine boilers  is  about  the  same,  namely,  about  10  square  feet  per 
horse  power. 

VII.  Area  of  Chimney. — RULE  1. —  To  find  the  area  of  chimney. 
— Multiply  the  number  of  nominal  horse  power  by  10*23;  the  pro- 
duct is  the  area  of  chimney  in  square  inches. 

Required  the  area  of  the  chimney  for  an  engine  of  400  nominal 
horse  power. 

In  this  example  we  have,  according  to  the  rule, 
area  of  chimney  in  square  inches  =  400  X  10-23  =  4092. 

We  may  also  find  a  rule  for  connecting  together  the  area  of  the 
chimney,  the  length  of  the  stroke,  and  the  diameter  of  the  cylinder. 

RULE  2. —  To  find  the  area  of  the  chimney. — Multiply  the  square 
of  the  diameter  expressed  in  inches  by  the  cube  root  of  the  stroke 
expressed  in  feet ;  divide  the  product  by  the  number  5 ;  the  quo- 
tient expresses  the  number  of  square  inches  in  the  area  of  chimney. 

Required  the  area  of  the  chimney  for  an  engine  whose  stroke  =* 
8  feet,  and  diameter  of  cylinder  =  50  inches. 

We  have  in  this  example  from  the  rule, 

502  x  ^8       2500  x  2 

area  of  chimney  m  square  inches  = ^ =  •= =• 

1000. 

w2  17 


\ 

258  THE   PRACTICAL   MODEL   CALCULATOR. 

To  work  this  example  according  to  the  first  rule,  we  find,  that 
the  nominal  horse  power  of  this  engine  is  104-6  :  hence, 

area  of  chimney  in  square  inches  =  104-6  X  10-23  =  1070. 

The  latter  value  is  greater  than  the  former  one  by  70  inches. 
This  difference  arises  from  our  taking  too  great  a  divisor  in  Rule  2. 
Either  of  the  values,  however,  is  near  enough  for  all  practical 
purposes. 

VIII.  Water  in  Boiler. — The  quantity  of  water  in  the  boiler 
differs  not  only  for  different  boilers,  but  differs  even  for  the  same 
boiler  at  different  times.     It  may  be  useful,  however,  to  know  the 
average  quantity  of  water  in  the  boiler  for  an  engine  of  a  given 
horse  power. 

RULE  1. — To  determine  the  average  quantity  of  water  in  the 
boiler. — Multiply  the  number  of  horse  power  by  5 ;  the  product 
expresses  the  cubic  feet  of  water  usually  in  the  boiler. 

This  rule  may  be  so  modified  as  to  make  it  depend  upon  the 
stroke  and  diameter  of  the  cylinder  of  engine. 

RULE  2. —  To  determine  the  cubic  feet  of  water  usually  in  the 
boiler. — Multiply  together  the  cube  root  of  the  stroke  in  feet,  the 
square  of  the  diameter  of  the  cylinder  in  inches,  and  the  number  5  ; 
divide  the  continual  product  by  47  ;  the  quotient  expresses  the  cu- 
bic feet  of  water  usually  in  the  boiler. 

Required  the  usual  quantity  of  water  in  the  boilers  of  an  engine 
whose  stroke  =  8  feet,  and  diameter  of  cylinder  50  inches. 

Here  we  have  from  the  rule, 

5  x  502  x  ^8"     5  x  2500  x  2 
cubic  feet  of  water  in  boiler  =  -     — ^ —  — 47 — 

25000 
=  — j= —  =  532  nearly. 

The  engine,  with  the  dimensions  we  have  specified,  is  of  106-4 
nominal  horse  power.  Hence,  according  to  Rule  1, 

cubic  feet  of  water  in  boiler  =  106-4  X  5  =  532. 

IX.  Area  of  Water  Level. — RULE  1. — To  find  the  area  of  water 
level. — The  area  of  water  level  contains  the  same  number  of  square 
feet  as  there  are  units  in  the  number  expressing  the  nominal  horse 
power  of  the  engine. 

Required  the  area  of  water  level  for  an  engine  of  200  nominal 
horse  power.  According  to  the  rule,  the  answer  is  200  square 
feet. 

We  add  a  rule  for  finding  the  area  of  water  level  when  the  di- 
ameter of  cylinder  and  the  length  of  stroke  is  given. 

RULE  2. —  To  find  the  area  of  water  level. — Multiply  the  square 
of  the  diameter  in  inches  by  the  cube  root  of  the  stroke  in  feet ; 
divide  the  product  by  47 ;  the  quotient  expresses  the  number  of 
square  feet  in  the  area  of  water  level. 

Required  the  area  of  the  water  level  for  an  engine  whose  stroke 
is  8  feet,  and  diameter  of  cylinder  50  inches. 


THE    STEAM  ENGINE.  259 

In  this  case,  according  to  the  rule, 


502  x 
area  of  water  level  m  square  feet  =  ---  j^  --  =  lOb. 

X.  Steam  Room.  —  It  is  obvious  that  the  steam  room,  like  the 
quantity  of  water,  is  an  extremely  variable  quantity,  differing,  not 
only  for  different  boilers,  but  even  in  the  same  boiler  at  different 
times.  It  is  desirable,  however,  to  know  the  content  of  that  part 
of  the  boiler  usually  filled  with  steam. 

RULE  1.  —  To  determine  the  average  quantity  of  steam  room.  — 
Multiply  the  number  expressing  the  nominal  horse  power  by  3  ; 
the  product  expresses  the  average  number  of  cubic  feet  of  steam 
room. 

Required  the  average  capacity  of  steam  room  for  an  engine  of 
460  nominal  horse  power. 

According  to  the  rule, 

Average  capacity  of  steam  room  =  460  X  3  cubic  feet  =  1380 
cubic  feet. 

This  rule  may  be  so  modified  as  to  apply  when  the  length  of 
stroke  and  diameter  of  cylinder  are  given. 

RULE  2.  —  Multiply  the  square  of  the  diameter  of  the  cylinder 
in  inches  by  the  cube  root  of  the  stroke  in  feet  ;  divide  the  product 
by  15  ;  the  quotient  expresses  the  number  of  cubic  feet  of  steam 
room.  « 

Required  the  average  capacity  of  steam  room  for  an  engine  whose 
stroke  is  8  feet,  and  diameter  of  cylinder  5  inches. 

In  this  case,  according  to  the  rule, 

502  x  &8       2500  x  2       5000 

Steam  room  in  cubic  feet  =  -  =-=  -  =  -  =-=  -  =    .,  -    = 

15  15  It) 

333J. 

We  find  that  the  nominal  horse  power  of  this  engine  is  106-4  ; 
hence,  according  to  Rule  1, 

average  steam  room  in  cubic  feet  =  106-4  X  3  =  320  nearly. 

Before  leaving  these  rules,  we  would  again  repeat  that  they  ought 
not  to  be  considered  as  rules  founded  upon  considerations  for  giving 
the  maximum  effect  from  the  combustion  of  a  given  amount  of  fuel  ; 
and  consequently  the  engineer  ought  not  to  consider  them  as  inva- 
riable, but  merely  to  be  followed  as  far  as  circumstances  will  per- 
mit. We  give  them,  indeed,  as  the  medium  value  of  the  very  va- 
riable practice  of  several  well-known  constructors  ;  consequently, 
although  the  proportions  given  by  the  rules  may  not  be  the  best 
possible  for  producing  the  most  useful  effect,  still  the  engineer  who 
is  guided  by  them  is  sure  not  to  be  very  far  from  the  common  prac- 
tice of  most  of  our  best  engineers.  It  has  often  been  lamented  that 
the  methods  used  by  different  engine  makers  for  estimating  the 
nominal  powers  of  their  engines  have  been  so  various  that  we  can 
form  no  real  estimate  of  the  dimensions  of  the  engine,  from  its  re- 
puted nominal  horse  power,  unless  we  know  its  maker  ;  but  the 


260  THE   PRACTICAL   MODEL   CALCULATOR. 

same  confusion  exists,  also,  to  some  extent,  in  the  construction  of 
boilers.  Indeed,  many  things  may  be  mentioned,  which  have 
hitherto  operated  as  a  barrier  to  the  practical  application  of  any- 
standard  of  engine  power  for  proportioning  the  different  parts  of 
the  boiler  and  furnace.  The  magnitude  of  furnace  and  the  extent 
of  heating  surface  necessary  to  produce  any  required  rate  of  eva- 
poration in  the  boiler  are  indeed  known,  yet  each  engine-maker 
has  his  own  rule  in  these  matters,  and  which  he  seems  to  think  pre- 
ferable to  all  others,  and  there  are  various  circumstances  influ- 
encing the  result  which  render  facts  incomparable  unless  those  cir- 
cumstances are  the  same.  Thus  the  circumstances  that  govern  the 
rate  of  evaporation,  as  influenced  by  different  degrees  of  draught, 
may  be  regarded  as  but  imperfectly  known.  And,  supposing  the 
difficulty  of  ascertaining  this  rate  of  evaporation  were  surAiounted, 
there  would  still  remain  some  difficulty  in  ascertaining  the  amount 
of  power  absorbed  by  the  condensation  of  the  steam  on  its  passage 
to  the  cylinder — the  imperfect  condensation  of  the  same  steam  after 
it  has  worked  the  piston — the  friction  of  the  various  moving  parts 
of  the  machinery — and,  especially,  the  difference  of  effect  of  these 
losses  of  power  in  engines  constructed  on  different  scales  of  magni- 
tude. Practice  must  often  vary,  to  a  certain  extent,  in  the  con- 
struction of  the  different  parts  of  the  boiler  and  furnace  of  an  en- 
gine ;  for,  independently  of  the  difficulty  of  solving  the  general 
problem  in  engineering,  the  determination  of  the  maximum  effect 
with  the  minimum  of  means,  practice  would  still  require  to  vary 
according  as  in  any  particular  case  the  desired  minimum  of  means 
was  that  of  weight,  or  bulk,  or  expense  of  material.  Again,  in  es- 
timating the  proper  proportions  for  a  boiler  and  its  appendages, 
reference  ought  to  be  made  to  the  distinction  between  the  "  power" 
or  "  effect"  of  the  boiler,  and  its  "  duty."  This  is  a  distinction  to 
be  considered  also  in  the  engine  itself.  The  power  of  an  engine 
has  reference  to  the  time  it  takes  to  produce  a  certain  mechanical 
effect  without  reference  to  the  amount  of  fuel  consumed ;  and,  on 
the  other  hand,  the  duty  of  an  engine  has  reference  to  the  amount 
of  mechanical  effect  produced  by  a  certain  consumption  of  fuel,  and 
is  independent  of  the  time  it  takes  to  produce  that  effect.  In  ex- 
pressing the  duty  of  engines,  it  would  have  prevented  much  need- 
less confusion  if  the  duty  of  the  boiler  had  been  entirely  separated 
from  that  of  the  engine,  as,  indeed,  they  are  two  very  distinct 
things.  The  duty  performed  by  ordinary  land  rotative  steam  en- 
gines is — 

One  horse  power  exerted  by  10  Ibs.  of  fuel  an  hour ;  or, 
Quarter  of  a  million  of  Ibs.  raised  1  foot  high  by  1  Ib.  of  coal ;  or, 
Twenty  millions  of  Ibs.  raised  one  foot  by  each  bushel  of  coals. 
Though  in  the  best  class  of  rotative  engines  the  consumption  is 
not  above  half  of  this  amount. 

The  constant  aim  of  different  engine  makers  is  to  increase  the 
amount  of  the  duty  ;  that  is,  to  make  10  Ibs.  of  fuel  exert  a  greater 
effect  than  one  horse  power ;  or,  in  other  words,  to  make  1  Ib.  of 


THE   STEAM   ENGINE. 


261 


coal  raise  more  than  a  quarter  of  a  million  of  Ibs.  one  foot  high. 
To  a  great  extent  they  have  been  successful  in  this.  They  have 
caused  5  Ibs.  of  coal  to  exert  the  force  of  one  horse  power,  and  even 
in  some  cases  as  little  as  3J  Ibs. ;  but  in  these  latter  cases  the 
economy  is  due  chiefly  to  expansive  action.  In  some  of  the  engines, 
however,  working  with  a  consumption  of  10  Ibs.  of  coal  per  nominal 
horse  hower  per  hour,  the  power  really  exerted  amounts  to  much 
more  than  that  represented  by  33,000  Ibs.  lifted  one  foot  high  in 
the  minute  for  each  horse  power.  Some  engines  lift  56,000  Ibs. 
one  foot  high  in  the  minute  by  each  horse  power,  with  a  consump- 
tion of  10  Ibs.  of  coal  per  horse  power  per  hour ;  and  even  this 
performance  has  been  somewhat  exceeded  without  a  recourse  to  ex- 
pansive action.  In  all  modern  engines  the  actual  performance 
much  exceeds  the  nominal  power ;  and  reference  must  be  had  to 
this  circumstance  in  contrasting  the  duty  of  different  engines. 

MECHANICAL   POWER   OP   STEAM. 

We  may  here  give  a  table  of  some  of  the  properties  of  steam, 
and  of  its  mechanical  effects  at  different  pressures.  This  table  may 
help  to  solve  many  problems  respecting  the  mechanical  effect  of 
steam,  usually  requiring  much  laborious  calculation. 


MECHANICAL  EFFECT  IN  HORSE  POWER  OF  1  LB. 

PRESSURES. 

Tern- 

Weight 

Velocity 

OF  STEAM. 

^d"6 

Cubic 

Without  Condensation. 

Condensation. 

I&MI. 

Foot 
Steam. 

Exit. 

Expansion. 

Expansion. 

Atmo- 
sphere. 

Lbs.  per 
Sq.  Inch. 

0 

i 

I 

i 

0 

\ 

* 

i 

1-00 

14-70 

212-00 

0-0364 

0 

0 

32-4 

95-2 

170-5 

913 

150-1 

178-6 

194-6 

1-25 

18-38 

223-88 

0-0440 

873 

21-5 

10-1 

32-3 

87-4 

95-9 

158-7 

190-6 

209-9 

1-50 

22-05 

234-32 

0-0529 

1135 

36-4 

39-3 

10-8 

30-6 

99-3 

165-2 

199-6 

221-1 

1-75 

25-72 

242-78 

0-0609 

1295 

47-41    60-8 

42-5 

11-1 

102-0 

170-0 

206-2 

229-5 

2-00 

29-40 

250-79 

0-0688 

1407 

55-9      77-5 

67-0 

43-2 

104-3 

174-2 

212-0 

236-5 

2-25 

33-08 

257-90 

0-0766 

1491 

62-8 

90-9 

86-5 

68-8 

106-2 

177-7 

"216-7 

242-4 

2-50 

36-75 

263-93 

0-0344 

1556 

68-4 

101-8 

102-4 

89-6 

107-7    180-5 

220-5 

247-1 

2-75 

40-42 

269-87 

0-0921 

1608 

73-1 

111-0 

115-8 

107-1 

109-3  1  183-2 

224-2 

251-6 

3-00 

44-10 

275-00 

0-0998 

1652 

71-1 

118-8 

127-1 

121-9 

110-6  ;  185-4 

227-7 

255-2 

3-35 

47-78 

279-86 

0-1073 

1690 

80-7 

125-6 

137-1 

136-7 

111-7  j  187-6 

230-0 

258-7 

3-50 

51-45 

284-63 

0-1148 

1722 

83-8 

131-5 

145-6 

145-8 

112-7  i  189-4 

232-4 

261-6 

3-75 

5512 

288-66 

0-1225 

1750 

86-5 

1^6-8 

153-2 

155-6 

113-7  |  190-1 

234-7 

264-4 

4-00 

58-18 

292-91 

0-1298 

1774 

89-0 

141-5 

160 

164-5 

114-6 

192-8 

236-9 

267-0 

4-50 

66-15 

300-27 

0-1445 

1816 

93-2 

149-8 

171-5 

179-4 

116-2 

195-6 

240-5 

271-4 

5-00 

73-50 

307-94 

0-1590 

1850 

96-8 

156-5 

181  6 

192-0 

117-7 

198-3 

244-1 

275-6 

6-00 

88-20 

320-00 

0-1878 

1904 

102-5 

167-2 

196-5 

211-4 

120-2 

202-6 

249-7 

282-2 

7-00 

102-90 

331-56 

0-2159 

1945 

107-0 

1756 

208-4 

226-5 

122-4 

206-4 

254-6 

288-1 

8-00 

117-60 

340-83 

0-2436 

1978 

110-6 

182-4 

217-9 

238-4 

124-3 

209 

258-8 

292-1 

9-00 

132-30 

351-32 

0-2708 

2006 

113-7 

188-2 

225-9 

248-5 

1260 

212 

262-7 

293-6 

10-00 

147-00 

359-60 

0-2977 

2029 

116-3 

193-0 

232-5 

256-7 

127-5 

215 

266-0 

301-4 

12-50 

183-75 

377-42 

0-3642 

2074 

121-5 

202-5 

245-5 

273-0 

130-7 

220 

272-9 

309-5 

15-00 

220-50 

392-90 

0-4288 

2109 

125-7 

210-0 

255-6 

285-4 

133-4 

225 

278-9 

316-4 

17-50 

257-25 

406-40 

0-4924 

2136 

129-0 

216-0 

263-6 

295-2 

135-7 

229 

283-9 

322-2 

20 

294-00 

418-56 

0-5549 

2159 

131-8 

221-0 

270-3 

305-3 

137-8 

233 

2K8-3 

327-2 

25 

367-50 

429-34 

0-6775 

2196 

136-3 

229-1 

281-0 

316-2 

141-2 

238 

295-7 

335-8 

30 

441-00 

457-16 

0-7970 

2226 

140-0 

235-6 

289-5 

326-4 

144-2 

244 

302-0 

343-1 

It  is  quite  clear  that  although  there  is  no  theoretical  limit  to  the 
benefit  derivable  from  expansion,  there  must  be  a  limit  in  practice, 
arising  from  the  friction  incidental  to  the  use  of  very  large  cylin- 
ders, the  magnitude  of  the  deduction  due  to  uncondensed  vapour 
when  the  steam  is  of  a  very  low  pressure,  and  other  circumstances 
which  it  is  needless  to  relate.  It  is  clear,  too,  that  while  the  effi- 


262          THE  PRACTICAL  MODEL  CALCULATOR. 

ciency  of  the  steam  is  increased  by  expansive  action,  the  efficiency 
of  the  engine  is  diminished,  unless  the  pressure  of  the  steam  or  the 
speed  of  the  piston  be  increased  correspondingly ;  and  that  an  en- 
gine of  any  given  size  will  not  exert  the  same  power  if  made  to  ope- 
rate expansively  without  any  other  alteration  that  would  have  been 
realized  if  the  engine  had  been  worked  with  the  full  pressure  of  the 
steam.  In  the  Cornish  engines,  which  work  with  steam  of  40  Ibs. 
on  the  inch,  the  steam  is  cut  off  at  one-twelfth  of  the  stroke ;  but 
if  the  steam  were  cut  off  at  one-twelfth  of  the  stroke  in  engines  em- 
ploying a  very  low  pressure,  it  would  probably  be  found  that  there 
would  be  a  loss  rather  than  a  gain  from  carrying  the  expansion  so 
far,  as  the  benefit  might  be  more  than  neutralized  by  the  friction 
incidental  to  the  use  of  so  large  a  cylinder  as  would  be  necessary 
to  accomplish  this  expansion ;  and  .unless  the  vacuum  were  a  very 
good  one,  there  would  be  but  little  difference  between  the  pressure 
of  the  steam  at  the  end  of  the  stroke  and  the  pressure  of  the  va- 
pour in  the  condenser,  so  that  the  urging  force  might  not  at  that 
point  be  sufficient  to  overcome  the  friction.  In  practice,  therefore, 
in  particular  cases,  expansion  may  be  -carried  too  far,  though  theo- 
retically the  amount  of  the  benefit  increases  with  the  amount  of  the 
expansion. 

We  must  here  introduce  a  simple  practical  rule  to  enable  those 
who  may  not  be  familiar  with  mathematical  symbols  to  determine 
the  amount  of  benefit  due  to  any  particular  measure  of  expansion. 
When  expansion  is  performed  by  an  expansion  valve,  it  is  an  easy 
thing  to  ascertain  at  what  point  of  the  stroke  the  valve  is  shut  by 
the  cam,  and  where  expansion  is  performed  by  the  slide  valve  the 
amount  of  expansion  is  easily  determinable  when  the  lap  and  stroke 
of  the  valve  are  known. 

RULE. —  To  find  the  Increase  of  Efficiency  arising  from  working 
Steam  expansively. — Divide  the  total  length  of  the  stroke  by  the 
distance  (which  call  1)  through  which  the  piston  moves  before  the 
steam  is  cut  off.  The  hyperbolic  logarithm  of  the  whole  stroke  ex- 
pressed in  terms  of  the  part  of  the  stroke  performed  with  the  full 
pressure  of  steam,  represents  the  increase  of  efficiency  due  to  ex- 
pansion. 

Suppose  that  the  pressure  of  the  steam  working  an  engine  is  45 
Ibs.  on  the  square  inch  above  the  atmosphere,  and  that  the  steam 
is  cut  off  at  one-fourth  of  the  stroke';  what  is  the  increase  of  effi- 
ciency due  to  this  measure  of  expansion  ? 

If  one-fourth  be  reckoned  as  1,  then  four-fourths  must  be  taken 
as  4,  and  the  hyperbolic  logarithm  of  4  will  be  found  to  be  1-386, 
which  is  the  increase  of  efficiency.  The  total  efficiency  of  the  quan- 
tity of  steam  expended  during  a  stroke,  therefore,  which  without 
expansion  would  have  been  1,  becomes  2-386  when  expanded  into  4 
times  its  bulk,  or,  in  round  numbers,  2-4. 

Let  the  pressure  of  the  steam  be  the  same  as  in  the  last  example, 
and  let  the  steam  be  cut  off  at  half-stroke :  what,  then,  is  the  in- 
crease of  efficiency  ? 


THE   STEAM    ENGINE. 


263 


Here  half  the  stroke  is  to  be  reckoned  as  1,  and  the  whole  stroke 
has  therefore  to  be  reckoned  as  2.  The  hyperbolic  logarithm,  of  2 
is  -693,  which  is  the  increase  of  efficiency,  and  the  total  efficiency 
of  the  stroke  is  1-693,  or  1-7. 

We  may  here  give  a  table  to  illustrate  the  mechanical  effect  of 
steam  under  varying  circumstances.  The  table  shows  the  me- 


Total 
pressure 
iu  Ibs. 
per 
Square 
Inch. 

Correspond- 
ing Tem- 
perature. 

Volume  of  Steam 
compared  with 
Water. 

Mechanical 
effect  of 
Cubic  Inuh  of 
Water. 

Total 
pressure 
in  Ibs. 
per 
Square 
Inch. 

Correspond- 
ing Tem- 
perature. 

Volume  of 
Steam 
compared 
with  Water. 

Mechanical 
effect  of 
Cubic  Iiidi 
of  Water. 

1 

103 

•  20-868 

1739 

51 

284 

544 

2312 

2 

126 

10-874 

1812 

52 

286 

534 

2816 

3 

141 

7437 

1859 

53 

287 

525 

2320 

4 

152 

5685 

1895 

54 

288 

516 

2324 

5 

161 

4617 

1924 

55 

289 

508 

2327 

6 

169 

3897 

1948 

66 

290J 

500 

2331 

7 

176 

3376 

1969 

57 

292" 

492 

2335 

8 

182 

2983 

1989 

58 

293 

484 

2339 

9 

187 

2674 

2006 

59 

294 

477 

2343 

10 

192 

2426 

2022 

60 

296 

470 

2347 

11 

197 

2221 

2036 

61 

297 

463 

2351 

12 

201 

2050 

2050 

62 

298 

456 

2355 

13 

205 

1904 

2063 

63 

299 

449 

2359 

14 

209 

1778 

2074 

64 

300 

443 

2362 

15 

218 

1669 

2086 

65 

301 

437 

2365 

16 

216 

1573 

2097 

66 

302 

431 

2369 

17 

220 

1488 

2107 

67 

303 

425 

2372 

18 

223 

1411 

2117 

68 

304 

419 

2375 

19 

226 

1343 

2126 

.  69 

305 

414 

2378 

20 

228 

1281 

2135 

70 

306 

408 

2382 

21 

231 

1225 

2144 

71 

307 

403 

2385 

22 

234 

1174 

2152 

72 

308 

398 

2388 

23 

236 

1127 

2160 

73 

309 

393 

2391 

24 

239 

1084 

2168 

74 

310 

388 

2394 

25 

241 

1044 

2175 

75 

311 

383 

2397 

26 

243 

1007 

2182 

76 

312 

379 

2400 

27 

245 

973 

2189 

77 

313 

374 

2403 

28 

248 

941 

2196 

78 

314 

370 

2405 

29 

250 

-911 

2202 

79 

315 

366 

2408 

30 

252 

883 

2209 

80 

316 

362 

2411 

31 

254 

857 

2215 

81 

317 

358 

2414 

32 

255 

833 

2221 

82 

318 

354 

2417 

33 

257 

810 

2226 

83 

318 

350 

2419 

34 

259 

788 

2232 

84 

319 

346 

2422 

35 

261 

767 

2238 

85 

320 

342 

2425 

36 

263 

748 

2243 

86 

321 

339 

2427 

37 

264 

729 

2248 

87 

322 

335 

2430 

38 

266 

712 

2253 

88 

323 

332 

2432 

39 

267 

695 

2259 

89 

323 

328 

2435 

40 

269 

679 

2264 

90 

324 

325 

2438 

41 

271 

664 

2268 

91 

325 

322 

2440 

42 

272 

649 

2273 

92 

326 

319 

2443 

43 

274 

635 

2278 

93 

327 

316 

2445 

44 

275 

622 

2282 

94 

327 

313 

2448 

45 

276 

610 

2287 

95 

328 

310 

2450 

46 

278 

598 

2291 

96 

329 

307 

2453 

47 

279 

586 

2296 

97 

330 

304 

2455 

48 

280 

575 

2300 

98 

330 

301 

2457 

49 

282 

564 

2304 

99 

331 

298 

2460 

50 

283 

554 

2308 

100 

332 

295 

2462 

264  THE  PRACTICAL  MODEL  CALCULATOR. 

chanical  effect  of  the  steam  generated  from  a  cubic  inch  of  water. 
Our  formula  gives  the  effect  of  a  cubic  foot  of  water ;  but  it  can  be 
modified  to  give  the  effect  of  the  steam  of  a  cubic  inch  by  dividing 
by  1728.  In  this  mannel-  we  find,  for  the  mechanical  effect  of  the 
gteam  of  a  cubic  inch  of  water,  about  3  (459  +  t]  Ibs.  raised  one 
foot  high.  The  table  shows  that  the  mechanical  effect  increases 
with  the  temperature.  The  increase  is  very  rapid  for  temperatures 
below  212°  ;  but  for  temperatures  above  this  the  increase  is  less ; 
and  for  the  temperatures  used  in  practice  we  may  consider,  with- 
out any  material  error,  the  mechanical  effect  as  constant. 

INDICATOR. 

An  instrument  for  ascertaining  the  amount  of  the  pressure  of 
steam  and  the  state  of  the  vacuum  throughout  the  stroke  of  a  steam 
engine.  Fitzgerald  and  Neucumn  long  employed  an  instrument 
of  this  kind,  the  nature  of  which  was  for  a  long  time  not  generally 
known.  Boulton  and  Watt  used  an  instrument  acting  upon  the 
same  principle  and  equally  accurate  ;  but  much  more  portable.  In 
peculiarity  of  construction  it  is  simply  a  small  cylinder  truly  bored, 
and  into  which  a  piston  is  inserted  and  loaded  by  a  spring  of  suit- 
able elasticity  to  the  graduated  scale  thereon  attached. 

The  action  of  an  indicator  is  that  of  describing,  on  a  piece  of 
paper  attached,  a  diagram  or  figure  approximating  more  or  less  to 
that  of  a  rectangle,  varying  of  course  with  the  merits  or  demerits 
of  the  engine's  productive  effect.  The  breadth  or  height  of  the 
diagram  is  the  sum  of  the  force  of  the  steam  and  extent  of  the  va- 
cuum ;  the  length  being  the  amount  of  revolution  given  to  the  paper 
during  the  piston's  performance  of  its  stroke. 

To  render  the  indicator  applicable,  it  is  commonly  screwed  into 
the  cylinder  cover,  and  the  motion  to  the  paper  obtained  by  means 
of  a  sufficient  length  of  small  twine  attached  to  one  of  the  radius 
bars ;  but  such  application  cannot  always  be  conveniently  effected, 
more  especially  in  engines  on  the  marine  principle  ;  hence,  other 
parts  of  such  engines,  and  other  means  whereby  to  effect  a  proper 
degree  of  motion,  must  unavoidably  be  resorted  to.  In  those  of 
direct  action  the  crosshead  is  the  only  convenient  place  of  attach- 
ment ;  but  because  the  length  of  the  engine's  stroke  is  considerably 
more  than  the  movement  required  for  the  paper  on  the  indicator, 
it  is  necessary  to  introduce  a  pulley  and  axle,  by  which  means  the 
various  movements  are  qualified  to  suit  each  other. 

When  the  indicator  is  fixed  and  the  movement  for  the  paper  pro- 
perly adjusted,  allow  the  engine  to  make  a  few  revolutions  previous 
to  opening  the  cock ;  by  which  means  a  horizontal  line  will  be  de- 
scribed upon  the  paper  by  the  pencil  attached,  and  denominated 
the  atmospheric  line,  because  it  distinguishes  between  the  effect  of 
the  steam  and  that  of  the  vacuum.  Open  the  cock,  and  if  the  en- 
gine be  upon  the  descending  stroke,  the  steam  will  instantly  raise 
the  piston  of  the  indicator,  and,  by  the  motion  of  the  paper  with  the 
pencil  pressing  thereon,  the  top  side  of  the  diagram  will  be  formed. 


THE   STEAM   ENGINE. 


265 


10.8 


12 


^      12.8 


O     12.8 


7.6 


12.8 


7.6 


12.8 


7.6 


12.8 


At  the  termination  of  the  stroke  and  immediately  previous  to  its 
return,  the  piston  of  the  indicator  is  pressed  down  by  the  surround- 
ing atmosphere,  consequently  the  bottom  side  of  the  diagram  is  de- 
scribed, and  by  the  time  the  engine  is  about  to  make  another  de- 
scending stroke,  the  piston  of  the  indicator  is  where  it  first  started 
from,  the  diagram  being  completed ;  hence  is  delineated  the  mean 
elastic  action  of  the  steam  above  that  of  the  atmospheric  line,  and 
also  the  mean  extent  of  the  vacuum  underneath  it. 

But  in  order  to  elucidate  more  B  n 

clearly  by  example,  take  the  follow- 
ing diagram,  taken  from  a  marine 
engine,  the  steam  being  cut  off  after 
the  piston  had  passed  through  two- 
thirds  of  its  stroke,  the  graduated 
scale  on  the  indicator,  tenths  of  an 
inch,  as  shown  at  each  end  of  the 
diagram  annexed. 

Previous  to  the  cock  being 
opened,  the  atmospheric  line  AB 
was  formed,  and,  when  opened,  the 
pencil  was  instantly  raised  by  the 
action  of  the  steam  on  the  piston 
to  C,  or  what  is  generally  termed 
the  starting  corner ;  by  the  move- 
ment of  the  paper  and  at  the  ter- 
mination of  the  stroke  the  line  CD 
was  formed,  showing  the  force  of 
the  steam  and  extent  of  expansion  ; 
from  D  to  E  show  the  moments  of  A 

eduction ;  from  E  to  F  the  quality  of  the  vacuum ;  and  from  F  to 
A  the  lead  or  advance  of  the  valve  ;  thus  every  change  in  the  en- 
gine is  exhibited,  and  every  deviation  from  a  rectangle,  except  that 
of  expansion  and  lead  of  the  valve  show  the  extent  of  proportionate 
defect.  Expansion  produces  apparently  a  defective  diagram,  but 
in  reality  such  is  not  the  case,  because  the'  diminished  power  of  the 
engine  is  more  than  compensated  by  the  saving  in  steam.  Also 
the  lead  of  the  valve  produces  an  apparent  defect,  but  a  certain 
amount  must  be  given,  as  being  found  advantageous  to  the  working 
of  the  engine,  but  the  steam  and  eduction  corners  ought  to  be  as 
square  as  possible;  any  rounding  on  the  steam  corner  shows  a  de- 
fect from  want  of  lead ;  and  rounding  on  the  eduction  corner  that 
of  the  passages  or  apertures  being  too  small. 

RULE. —  To  compute  the  power  of  an  Engine  from  the  Indicator 
Diagram. — Divide  the  diagram  in  the  direction  of  its  length  into 
any  convenient  number  of  equal  parts,  through  which  draw  lines 
at  right  angles  to  the  atmospheric  line,  add  together  the  lengths  of 
all  the  spaces  taken  in  measurements  corresponding  with  the  scale 
on  the  indicator,  divide  the  sum  by  the  number  of  spaces,  and  the 


266  THE    PRACTICAL    MODEL    CALCULATOR. 

quotient  is  the  mean  effective   pressure  on  the  piston  in  Ibs.  per 
square  inch. 

Let  the  result  of  the  preceding  diagram  be  taken  as  an  example. 
Then,  the  whole  sum  of  vacuum  spaces  =  1220  -*-  10  =  12*2  Ibs. 
mean  effect  obtained  by  the  vacuum ;  and  in  a  similar  manner  the 
mean  effective  pressure  of  steam  is  found  to  be  6 -28  Ibs.,  hence  the 
total  effective  force  =  18*48  Ibs.  per  square  inch.  And  supposing 
2-5  Ibs.  per  square  inch  be  absorbed  by  friction,  What  is  the  actual 
power  of  the  engine,  the  cylinder's  diameter  being  32  inches,  and 
the  velocity  of  the  piston  226  feet  per  minute  ? 

18*48  —  2*5  =  15*98  Ibs.  per  square  inch  of  net  available  force. 

322  x  -7854  x  15-98  x  226 
Then  -  — ''SOOO —          =        norses  power. 

The  line  under  the  diagram  and  parallel  to  the  atmospheric  line 
is  xfths  distant,  and  represents  the  perfect  vacuum  line,  the  space 
between  showing  the  amount  of  force  with  which  the  uncondensed 
steam  or  vapour  resists  the  ascent  or  descent  of  the  piston  at  every 
part  of  the  stroke. 

As  the  mean  pressure  of  the  atmosphere  is  15  Ibs.  per  square 
inch,  and  the  mean  specific  gravity  of  mercury  13560,  or  2*037  cu- 
bic inches  equal  1  lb.,  it  will  of  course  rise  in  the  barometer  at- 
tached to  the  condenser  about  2  inches  for  every  lb.  effect  of  va- 
cuum, and  as  a  pure  vacuum  would  be  indicated  by  30  inches  of 
mercury,  the  distance  between  the  two  lines  shows  whether  there 
is  or  is  not  any  amount  of  defect,  as  sometimes  there  is  a  consider- 
able difference  in  extent  of  vacuum  in  the  cylinder  to  that  in  the 
condenser. 

To  estimate  by  means  of  an  indicator  the  amount  of  effective  power 
produced  by  a  steam  engine. — Multiply  the  area  of  the  piston  in 
square  inches  by  the  average  force  of  the  steam  in  Ibs.  and  by  the 
velocity  of  the  piston  in  feet  per  minute ;  divide  the  product  by 
33,000,  and  ^ths  of  the  quotient  equal  the  effective  power. 

Suppose  an  engine  wi^h  a  cylinder  of  37J  inches  diameter,  a 
stroke  of  7  feet,  and  making  17  revolutions  per  minute,  or  238  feet 
velocity,  and  the  average  indicated  pressure  of  the  steam  16*73  Ibs. 
per  square  inch ;  required  the  effective  power. 

Area  =  1104-4687  inches  X  16-73  Ibs.  X  238  feet 
~33000~ 

133-26  x  7 

— -JQ ==  93-282  horse  power. 

To  determine  the  proper  velocity  for  the  piston  of  a  steam  engine. — 
Multiply  the  logarithm  of  the  nth  part  of  the  stroke  at  which  the 
steam  is  cut  off  by  2*3,  and  to  the  product  of  which  add  -7.  Mul- 
tiply the  sum  by  the  distance  in  feet  the  piston  has  travelled  when 
the  steam  is  cut  off,  and  120  times  the  square  root  of  the  product 
equal  the  proper  velocity  for  the  piston  in  feet  per  minute. 


267 


WEIGHT   COMBINED  WITH  MASS,   VELOCITY,    FORCE,    AND 

WORK  DONE. 

CALCULATIONS  ON  THE  PRINCIPLE  OF  VIS  VIVA.  —  MATERIALS  EMPLOYED 
IN  THE  CONSTRUCTION  OF  MACHINES.  —  STRENGTH  OF  MATERIALS, 
THEIR  PROPERTIES.  -  TORSION,  DEFLEXION,  ELASTICITY,  TENACITIES, 
COMPRESSIONS,  ETC.  -  FRICTION  OF  REST  AND  OF  MOTION,  COEFFICIENTS 
OF  ALL  SORTS  OF  MOTION.  —  BANDS.  —  ROPES.  —  WHEELS.  -  HYDRAU- 
LICS. —  NEW  TABLES  FOR  THE  MOTION  AND  FRICTION  OF  WATER.  — 
WATER-WHEELS.  -  WINDMILLS,  ETC. 

1.  Suppose  a  body  resting  on  a  perfectly  smooth  table,  and,  when 
in  motion,  to  present  no  impediment  to  the  body  in  its  course,  but 
merely  to  counteract  the  force  of  gravity  upon  it  ;  if  this  body 
weighing  800  Ibs.  be  pressed  by  the  fortfe  of  30  Ibs.  acting  hori- 
zontally and  continuously,  the  motion  under  such  circumstances 
will  be  uniformly  accelerated  :  what  is  the  acceleration  ? 

30 

x  32-2  =  1-2075  feet  the  second. 


2.  What  force  is  necessary  to  move  the  above-mentioned  heavy 
body,  with  a  23  feet  acceleration,  under  the  same  circumstances  ? 

23 
g^2  x  80°  =  5714285  Ibs. 

The  second  of  these  examples  illustrates  the  principle  that  the 
force  which  impels  a  body  with  a  certain  acceleration  is  equal  to 
the  weight  of  the  body  multiplied  by  the  ratio  of  its  acceleration 
to  that  of  gravity.  The  first  illustrates  the  reverse,  namely,  the 
acceleration  with  which  a  body  is  moved  forward  with  a  given  force, 
is  equal  to  the  acceleration  of  gravity  multiplied  by  the  ratio  of  the 
force  to  the  weight. 

3.  A  railway  car,  weighing  1120  Ibs.,  moves  with  a  5  feet  velo- 
city upon  horizontal  rails,  which,  let  us  suppose,  offer  no  impedi- 
ment to  the  motion,  and  is  constantly  pushed  by  an  invariable 
force  of  50  Ibs.  during  20  seconds  :  with  what  velocity  is  it  moving 
at  the  end  of  the  20th  second,  or  at  the  beginning  of  the  21st 
second  ? 

50 
5  +  32-2  X  x  20  =  33-75,  the  velocity. 


4.  A  carriage,  circumstanced  as  in  the  last  question,  weighs  4000 
Ibs.  ;  its  initial  velocity  is  30  feet  the  second,  and  its  terminal  velo- 
city is  70  feet  :  with  which  force  is  the  body  impelled,  supposing  it 
to  be  in  motion  20  seconds  ? 


We  have  before  noticed  that  the  weight  (W),  divided  by  32-2,  or 
\  gives  the  mass;  that  is, 


268  THE    PRACTICAL    MODEL    CALCULATOR. 

Weight 
— —  =  mass, 

And,  force  =  mass  X  acceleration. 

5.  Suppose  a  railway  carriage,  weighing  6440  Ibs.,  moves  on  a 
horizontal  plane  offering  no  impediment,  and  is  uniformly  accele- 
rated 4  feet  the  second,  what  continuous  force  is  applied  ? 

6440 

32—  =  200  Ibs.  mass. 

200  X  4  =  800  Ibs.,  the  force  applied. 

By  the  four  succeeding  formulas,  all  questions  may  be  answered 
that  may  be  proposed  relative  to  the  rectilinear  motions  of  bodies 
by  a  constant  force. 

For  uniformly  accelerated  motions : 

F 
v~a  +  32-2^  x  f, 

F 
s  =  at  +  16-1  yj  X  t2. 

For  uniformly  retarded  motions  : 

F 
v  =  a  —  32-2  vy  x  t; 

F 
s  =  at  —  16-1  X  yf  X  t2', 

t  =  the  time  in  seconds,  W  =  the  weight  in  Ibs.,  F  =  the  force  in 
Ibs.,  a  =  the  initial  velocity,  and  v  =  the  terminal  velocity. 

6.  A  sleigh,  weighing  2000  Ibs.,  going  at  the  rate  of  20  feet  a 
second,  has  to  overcome  by  its  motion  a  friction  of  30  Ibs. :  what 
velocity  has  it  after  10  seconds,  and  what  distance  has  it  described  ? 

30 
20  -  32-2  x  200Q  x  10  =  15-17  feet  velocity. 

30 

20  x  10  -  16-1  x  200()  x  (10)2  =  175-85  feet,  distance  de- 
scribed. 

7.  In  order  to  find  the  mechanical  work  which  a  draught-horse 
performs  in  drawing  a  carriage,  an  instrument  called  a  dynamome- 
ter, or  measure  of  force,  is  thus  used  :  it  is  put  into  communication 
on  one  side  of  the  carriage,  and  on  the  other  with  the  traces  of  the 
horse,  and  the  force  is  observed  from  time  to  time.     Let  126  Ibs. 
be  the  initial  force;  after  40  feet  is  described,  let  130  Ibs.  be  the 
force  given  by  the  dynamometer ;  after  40  feet  more  is  described, 
let  129  Ibs.  be  the  force  ;  after  40  feet  more  is  passed  over,  let  140 
Ibs.  be  the  force  ;  and  let  the  next  two  spaces  of  40  feet  give  forces 
of  130  and  120  Ibs.  respectively.   What  is  the  mechanical  work  done  ? 

126  initial  force. 
120  terminal  force. 

2)246 

123  mean. 


WEIGHT   COMBINED    WITH    MASS,    VELOCITY,    ETC.  269 

123  +  130  -f  129  +  140  +  130 
— _  130-4 

130-4  x  40  x  5  =  26080  units  of  work. 

The  following  rule,  usually  given  to  find  the  areas  of  irregular 
figures,  may  be  applied  where  great  accuracy  is  required. 

RULE. — To  the  sum  of  the  first  and  last,  or  extreme  ordinates, 
add  four  times  the  sum  of  the  2d,  4th,  6th,  or  even  ordinates,  and 
twice  the  sum  of  the  3d,  5th,  7th,  &c.,  or  odd  ordinates,  not  includ- 
ing the  extreme  ones ;  the  result  multiplied  by  J  the  ordinates' 
equidistance  will  be  the  sum. 
126 
120 

246  sum  of  first  and  last. 

246  +  4  x  130  +lTx  129  +  4  x  140  +  2  x  130  =  1844.  x 

1844  x  40  \ 

— o —    -  =  24586-66  units  of  work  or  pounds  raised  one  foot 

high.  This  rule  of  equidistant  ordinates  is  of  great  use  in  the  art  of 
ship-building.  This  application  we  shall  introduce  in  the  proper 
place. 

8.  How  many  units  of  work  are  necessary  to  impart  to  a  carriage 
of  3000  Ibs.  weight,  resting  on  a  perfectly  smooth  railroad,  a  velo- 
city of  100  feet  ? 

S|L  x  3000  =  465838-2  units. 

A    X   oZi'Zi 

A  unit  of  work  is  that  labour  which  is  equal  to  the  raising  of  a 
pound  through  the  space  of  one  foot.  A  unit  of  work  is  done  when 
one  pound  pressure  is  exerted  through  a  space  of  one  foot,  no  matter 
in  what  direction  that  space  may  lie. 

Kane  Fitzgerald,  the  first  that  made  steam  turn  a  crank,  and 
patented  it,  and  the  fly-wheel  to  regulate  its  motion,  estimated  that 
a  horse  could  perform  33000  units  of  work  in  a  minute,  that  is, 
raise  33000  Ibs.  one  foot  high  in  a  minute.  To  perform  465838-2 
units  of  work  in  10  minutes  would  require  the  application  1-4116 
horse  power. 

9.  What  work  is  done  by  a  force,  acting  upon  another  carriage, 
under  the  same  circumstances,  weighing  5000  Ibs.,  which  transforms 
the  velocity  from  30  to  50  feet  ? 


64T4  =  13'990^?  the  neign*  due  to  30  feet  velocity. 

(50? 

g^  =  38-8043,  the  height  due  to  50  feet  velocity 

From    38-8043 
Take     13-9907 

24-8136 

5000 


124068-0000 

x2 


270  THE    PRACTICAL   MODEL   CALCULATOR. 

.-.  124068  are  the  units  of  work,  and  just  so  much  work  will  the 
carriage  perform  if  a  resistance  be  opposed  to  it,  and  it  be  gradu- 
ally brought  from  a  50  feet  velocity  to  a  30  feet  velocity. 

The  following  is  without  doubt  a  very  simple  formula,  but  the 
most  useful  one  in  mechanics  ;  by  it  we  have  solved  the  last  two 
questions  : 

Fs  =  (H  -  h)  W. 

This  simple  formula  involves  the  principle  technically  termed  the 
principle  of  vis  VIVA,  or  LIVING  FORCES.  H  is  the  height  due  to 

v2 
one  velocity,  say  v  or  H  =  ^~  and  A,  the  height  due  to  another  a, 

or  h  =  -Q-.     The  weight  of  the  mass  =  W  ;  the  force  F,  and  the 

/pace  8. 
To  express  this  principle  in  words,  we  may  say,  that  the  working 
power  (Fs)  which  a  mass  either  acquires  when  it  passes  from  a  lesser 
velocity  (a)  to  a  greater  velocity  (v\  or  produces  when  it  is  com- 
pelled to  pass  from  a  greater  velocity  (v)  into  a  less  (a),  is  always 
equal  to  the  product  of  the  weight  of  the  mass  and  the  difference 
of  the  heights  due  to  the  velocities. 

When  we  know  the  units  of  work,  and  the  distance  in  which  the 
change  of  velocity  goes  on,  the  force  is  easily  found  ;  and  when  the 
force  is  known,  the  distance  is  readily  determined.  Suppose,  in  the 
last  example,  that  the  change  of  velocity  from  30  to  50  feet  took 
place  in  a  distance  of  300  feet,  then 

124068 

=  413-56  Ibs.  =  F,  the  force  constantly  applied  during 


300  feet. 

10.  If  a  sleigh,  weighing  2000  Ibs.,  after  describing  a  distance  of 
250  feet,  has  completely  lost  a  velocity  of  100  feet,  what  constant 
resistance  does  the  friction  offer  ? 

Since  the  terminal  velocity  =  0,  the  height  due  to  it  =  0,  hence 


We  have  been  calculating  upon  the  principle  of  vis  viva;  but  the 
product  of  the  mass  and  the  square  of  the  velocity,  without  attach- 
ing to  it  any  definite  idea,  is  termed  the  vis  viva,  or  living  force. 

11.  A  body  weighing  2300  Ibs.  moves  with  a  velocity  of  20  feet 
the  second,  required  the  vis  viva  f 

||^  =  7142857  Ibs.,  mass. 

71-42857  X  (20)2  =  28571-428,  the  amount  of  vis  viva. 

Hence,  if  a  mass  enters  from  a  velocity  a,  into  another  v,  the 
unit  of  work  done  is  equal  to  half  the  difference  of  the  vis  viva,  at 
the  commencement  and  end  of  the  change  of  velocity. 

For  if  the  mass  be  put  =  M,  and  W  the  weight, 


STRENGTH   OF   MATERIALS.  271 

W  Wa2 

Then  M  ==  —  ,  and  the  vis  viva  to  velocity  a  —  Ma2  =  -—  ; 

9  y 

Wv2 

and  the  vis  viva  to  velocity  v  =  Mv2  =  -  .  ' 


2 

Q—  and  or,  give  the  heights  due  to  the  velocities  v  and  «,  respec- 

tively.    The  useful  formula 

Fs  =  (H  -  h)  W, 

before  given,  page  270,  may  be  applied  to  variable  as  well  as  to 
constant  forces,  if,  instead  of  the  constant  force  F,  the  mean  value 
of  the  force  be  applied. 


STRENGTH  OF  MATERIALS. 

ON   MATERIAL  EMPLOYED   IN    THE    CONSTRUCTION   OF   MACHINES. 

IN  theoretical  mechanics,  we  deal  with  imaginary  quantities,  which 
are  perfect  in  all  their  properties ;  they  are  perfectly  hard,  and 
perfectly  elastic  ;  devoid  of  weight  in  statics  and  of  friction  in  dy- 
namics. In  practical  mechanics,  we  deal  with  real  material  objects, 
among  which  we  find  none  which  are  perfectly  hard,  and  none,  ex- 
cept gaseous  bodies,  which  are  perfectly  elastic ;  all  have  weight, 
and  experience  resistance  in  dynamical  action.  Practical  mechanics 
is  the  science  of  automatic  labour,  and  its  objects  are  machines  and 
their  applications  to  the  transmission,  modification,  and  regulation 
of  motive  power.  In  this  it  takes  as  a  basis  the  theoretical  deduc- 
tions of  pure  mechanics,  but  superadds  to  the  formulae  of  the  ma- 
thematician a  multitude  of  facts  deduced  from  observation,  and  ex- 
perimentally elaborates  a  new  code  of  laws  suited  to  the  varied  con- 
ditions to  be  fulfilled  in  the  economy  of  the  industrial  arts. 

In  reference  to  the  structure  of  machines,  it  is  to  be  observed 
that  however  simple  or  complex  the  machine  may  be,  it  is  of  im- 
portance that  its  parts  combine  lightness  with  strength,  and  rigidity 
with  uniformity  of  action ;  and  that  it  communicates  the  power 
without  shocks  and  sudden  changes  of  motion,  by  which  the  passive 
resistances  may  be  increased  and  the  effect  of  the  engine  dimi- 
nished. 

To  adjust  properly  the  disposition  and  arrangement  of  the  indi- 
vidual members  of  a  machine,  implies  an  exact  knowledge  and  esti- 
mate of  the  amount  of  strain  to  which  they  are  respectively  subject 
in  the  working  of  the  machine ;  and  this  skill,  when  exercised  in 
conjunction  with  an  intimate  acquaintance  with  the  nature  of  the 
materials  of  which  the  parts  are  themselves  composed,  must  con- 
tribute to  the  production  of  a  machine  possessing  the  highest  amount 
of  capability  attainable  with  the  given  conditions. 

Materials. — The  material  most  commonly  employed  in  the  con- 


272          THE  PRACTICAL  MODEL  CALCULATOR. 

struction  of  machinery  is  iron,  in  the  two  states  of  cast  and  wrought 
or  forged  iron  ;  and  of  these,  there  are  several  varieties  of  quality. 
It  becomes  therefore  a  problem  of  much  practical  importance  to 
determine,  at  least  approximately,  the  capabilities  of  the  particular 
material  employed,  to  resist  permanent  alteration  in  the  directions 
in  which  they  are  subjected  to  strain  in  the  reception  and  trans- 
mission of  the  motive  power. 

To  indicate  briefly  the  fundamental  conditions  which  determine 
the  capability  of  a  given  weight  and  form  of  material  to  resist  a 
given  force,  it  must,  in  the  first  place,  be  observed,  that  rupture 
may  take  place  either  by  tension  or  by  compression  in  the  direc- 
tion of  the  length.  To  the  former  condition  of  strain  is  opposed 
the  tenacity  of  the  material ;  to  the  other  is  opposed  the  resistance 
to  the  crushing  of  its  substance.  Rupture,  by  transverse  strain,  is 
opposed  both  by  the  tenacity  of  the  material  and  its  capability  to 
withstand  compression  together  of  its  particles.  Lastly,  the  bar 
may  be  ruptured  by  torsion.  Mr.  Oliver  Byrne,  the  author  of  the 
present  work,  in  his  New  Theory  of  the  Strength  of  Materials  has 
pointed  out  new  elements  of  much  importance. 

The  capabilities  of  a  material  to  resist  extension  and  compression 
are  often  different.  Thus,  the  soft  gray  variety  of  cast  iron  offers 
a  greater  resistance  to  a  force  of  extension  than  the  white  variety 
in  a  ratio  of  nearly  eight  to  five ;  but  the  last  offers  the  greatest 
resistance  to  a  compressing  force. 

The  resistance  of  cast  iron  to  rupture  by  extension  varies  from 
6  to  9  tons  upon  the  square  inch  ;  and  that  to  rupture  by  compres- 
sion, from  36  to  65  tons.  The  resistance  to  extension  of  the  best 
forged  iron  may  be  reckoned  at  25  tons  per  inch ;  but  the  corre- 
sponding resistance  to  compression,  although  not  satisfactorily  ascer- 
tained, is  generally  considered  to  be  greatly  less  than  that  of  cast 
iron.  Roudelet  makes  it  31J  tons  on  the  square  inch.  Cast  iron 
(and  even  wood)  is  therefore  to  be  preferred  for  vertical  supports. 

The  forces  resisting  rupture  are  as  the  areas  of  the  sections  of 
rupture,  the  material  being  the  same ;  this  principle  holds  not  only 
in  respect  of  iron,  but  also  of  wood.  Many  inquiries  have  been  in- 
stituted to  determine  the  commonly  received  principle,  that  the 
strength  of  rectangular  beams  of  the  same  width  to  resist  rupture 
by  transverse  strain  is  as  the  squares  of  the  depths  of  the  beams. 

In  these  respects  the  experiments,  although  valuable  on  account 
of  their  extent  and  the  care  with  which  they  were  conducted,  pos- 
sess little  novelty ;  but  in  directing  attention  to  the  elastic  proper- 
ties of  the  materials  experimented  upon,  it  was  found  that  the  re- 
ceived doctrine  of  relation  between  the  limit  of  elasticity  and  weight 
requires  modification.  The  common  assumption  is,  that  the  de- 
struction of  the  elastic  properties  of  a  material,  that  is,  the  dis- 
placement beyond  the  elastic  limit,  does  not  manifest  itself  until 
the  load  exceeds  one-third  of  the  breaking  weight.  It  was  found, 
however,  on  the  contrary,  that  its  effect  was  produced  and  mani- 
fested in  a  permanent  set  of  the  material  when  the  load  did  not  ex- 


STRENGTH    OF   MATERIALS.  273 

ceed  one-sixteenth  of  that  necessary  to  produce  rupture.  Thus  a 
bar  of  one  inch  square,  supported  between  props  4J  feet  apart,  did 
not  break  till  loaded  with  496  Ibs.  but  showed  a  permanent  deflec- 
tion or  set  when  loaded  with  .16  Ibs.  In  other  cases,  loads  of  7  Ibs. 
and  14  Ibs.  were  found  to  produce  permanent  sets  when  the  break- 
ing weights  were  respectively  364  Ibs.  and  1120  Ibs.  These  sets 
were  therefore  given  by  ^d  and  ^th  of  the  breaking  weights. 

Since  these  results  were  obtained,  it  has  been  found  that  time 
and  the  weight  of  the  material  itself  are  sufficient  to  effect  a  per- 
manent deflection  in  a  beam  supported  between  props,  so  that  there 
would  seem  to  be  no  such  limits  in  respect  to  transverse  strain  as 
those  known  by  the  name  of  elastic  limits,  and  consequently  the 
principle  of  loading  a  beam  within  the  elastic  limit  has  no  founda- 
tion in  practice.  The  beam  yields  continually  to  the  load,  but  with 
an  exceedingly  slow  progression,  until  the  load  approximates  to  the 
breaking  weight,  when  rupture  speedily  succeeds  to  a  rapid  deflection. 

As  respects  the  effect  of  tension  and  compression  by  transverse 
strain,  it  was  ascertained  by  a  very  ingenious  experiment  that  equal 
loads  produced  equal  deflections  in  both  cases. 

Another  most  important  principle  developed  by  experiments,  is 
that  respecting  the  compression  of  supporting  columns  of  different 
heights.  When  the  height  of  the  column  exceeded  a  certain  limit, 
it  was  found  that  the  crushing  force  became  constant,  and  did  not 
ii^crease  as  the  height  of  the  column  increased,  until  it  reached 
another  limit  at  which  it  began  to  yield,  not  strictly  by  crushing, 
but  by  the  bending  of  the  material.  The  first  limit  was  found  to  be 
a  height  of  little  less  than  three  times  the  radius  of  the  column ; 
and  the  second  double  that  height,  or  about  six  times  the  radius  of 
the  column.  In  columns  of  different  heights  between  these  limits, 
having  equal  diameters,  the  force  producing  rupture  by  compression 
was  nearly  constant.  When  the  column  was  less  than  the  lower 
limit,  the  crushing  force  became  greater,  and  when  it  was  greater 
than  the  higher  limit,  the  crushing  force  became  less.  It  was  fur- 
ther found  that  in  all  cases,  where  the  height  of  the  column  was 
exactly  above  the  limits  of  three  times  the  radius,  the  section  of 
rupture  was  a  plane  inclined  at  nearly  the  same  constant  angle  of 
55  degrees  to  the  axis  of  the  column.  These  facts  mutually  ex- 
plain each  other ;  for  in  every  height  of  column  above  the  limit, 
the  section  of  rupture  being  a  plane  at  the  same  angle  to  the  axis 
of  the  column,  must  of  necessity  be  a  plane  of  the  same  size,  and 
therefore  in  each  case  the  cohesion  of  the  same  number  of  particles 
must  be  overcome  in  producing  rupture.  And  further,  the  same 
number  of  particles  being  to  be  overcome  under  the  same  circum- 
stances for  every  different  height,  the  same  force  will  be  required 
to  overcome  that  amount  of  cohesion,  until  at  double  the  height 
(three  diameters)  the  column  begins  to  bend  under  its  load.  This 
height  being  surpassed,  it  follows  that  a  pressure  which  becomes 
continually  less  as  the  length  of  the  column  is  increased,  will  be 
sufficient  to  break  it. 

18 


274 


THE  PRACTICAL  MODEL  CALCULATOR. 


This  property,  moreover,  is  not  confined  to  cast  iron ;  the  ex- 
periments of  M.  Rondelet  show  that  with  columns  of  wrought  iron, 
wood,  and  stone,  similar  results  are  obtained. 

From  these  facts  then,  it  appears  that  if  supporting  columns  be 
taken  of  different  diameters,  and  of  heights  so  great  as  not  to  allow 
of  their  bending,  yet  sufficiently  high  to  allow  of  a  complete  sepa- 
ration of  the  planes  of  fracture,  that  is,  of  heights  intermediate  to 
three  times  and  six  times  their  radius,  then  will  their  strengths  be 
as  the  number  of  particles  in  their  planes  of  fracture ;  and  the 
planes  of  fracture  being  inclined  at  equal  angles  to  the  axes  of  the 
columns,  their  areas  will  be  as  the  transverse  sections  of  the  co- 
lumns, and  consequently  the  strengths  of  the  columns  will  be  as 
their  transverse  sections  respectively.  Taking  the  mean  of  three 
experiments  upon  a  column  J  inch  diameter,  the  crushing  force  was 
6426  Ibs. ;  whilst  the  mean  of  four  experiments,  conducted  in  ex- 
actly the  same  manner,  upon  a  column  of  f  of  an  inch  diameter, 
gave  14542  Ibs.  The  diameters  of  the  columns  being  2  to  3,  the 
areas  of  transverse  section  were  therefore  4  to  9,  which  is  very 
nearly  the  ratio  of  the  crushing  weights. 

When  the  length  of  the  column  is  so  great  that  its  fracture  is 
produced  wholly  by  bending  of  its  material,  the  limit  has  been  fixed 
for  columns  of  cast  iron,  at  30  times  the  diameter  when  the  ends 
are  flat,  and  15  times  the  diameter  when  the  ends  are  rounded.  In 
shorter  columns,  fracture  takes  place  partly  by  crushing  and  partly 
by  bending  of  the  material.  When  the  column  is  enlarged  in  the 
middle  of  its  length  from  one  and  a  half  to  two  times  the  diameter 
of  the  ends,  the  strength  was  found  by  the  same  experimenter  to  be 
greater  by  one-seventh  than  in  solid  columns  containing  the  same 
quantity  of  iron,  in  the  same  length,  with  their  extremities  rounded ; 
and  stronger  by  an  eighth  or  a  ninth  when  the  extremities  were  flat 
and  rendered  immovable  by  disks. 

The  following  formulas  give  the  absolute  strength  of  cylindrical 
columns  to  sustain  pressure  in  the  direction  of  their  length.  In 
these  formulas 

D  ==  the  external  diameter  of  the  column  in  inches. 
d  =  the  internal  diameter  of  hollow  columns  in  inches. 
L  =  the  length  of  the  column  in  feet. 
W  =  the  breaking  weight  in  tons. 


Character  of  the  column. 

Length  of  the  column  exceeding  15 
times  its  diameter. 

Length  of  the  column  exceedi 
times  its  diameter. 

ng  30 

Solid    cylindrical    co-  \ 
lumn  of  cast  iron,        / 
Hollow  cylindrical  co-  ") 
lumn  of  cast  iron,        / 
Solid    cylindrical    co-  ) 
lumn  of  wrought  iron,  / 

Both  ends  rounded. 

D3-76 

W1  i\  -0 

W  r= 
W  = 
W  = 

Both  ends  flat. 
AA.1  A 

3-55 

D*™  d*-"> 

\V          1  ^ 

T)  8  s*  0 

D3-I6 

W  =  42-8^— 

133-75  J? 
LI 

For  shorter  columns,  if  W  represent  the  weight  in  tons  which 
would  break  the  column  by  bending  alone,  as  given  by  the  preced- 


STRENGTH    OF   MATERIALS.  275 

ing  formulas,  and  W"  the  weight  in  tons  which  would  crush  the  co- 
lumn without  bending  it,  as  determined  from  the  subjoined  table, 
then  the  absolute  breaking  weight  of  the  column  W,  is  represented 
in  tons  by  the  formula, 

_  W'  X  W" 
W  =~  W  +  W" 
These  rules  require  the  use  of  logarithms  in  their  applications. 

When  a  beam  is  deflected  by  transverse  strain,  the  material  on 
that  side  of  it  on  which  it  sustains  the  strain  is  compressed,  and  the 
material  on  the  opposite  side  is  extended.  The  imaginary  surface 
at  which  the  compression  terminates  and  the  extension  begins — at 
which  there  is  supposed  to  be  neither  extension  nor  compression — 
is  termed  the  neutral  axis  of  the  beam.  What  constitutes  the 
strength  of  a  beam  is  its  resistance  to  compression  on  the  one  side 
and  to  extension  on  the  other  side  of  that  axis — the  forces  acting 
about  the  line  of  axis  like  antagonist  force  at  the  two  extremities 
of  a  lever,  so  that  if  either  of  them  yield,  the  beam  will  be  broken. 
It  becomes,  however,  a  question  of  importance  to  determine  the  re- 
lation of  these  forces ;  in  other  words,  to  determine  whether  the 
beam  of  given  form  and  material  will  yield  first  to  compression  or 
to  extension.  This  point  is  settled  by  reference  to  the  columns  of 
the  subsequent  table,  page  280,  in  which  it  will  be  observed  that  the* 
metals  require  a  much  greater  force  to  crush  them  than  to  tear  them 
asunder,  and  that  the  woods  require  a  much  smaller  force. 

There  is  also  another  consideration  which  must  not  be  overlooked. 
Bearing  in  mind  the  condition  of  antagonism  of  the  forces,  it  is  ob- 
vious, that  the  further  these  forces  are  placed  from  the  neutral 
axis,  that  is,  from  the  fulcrum  of  their  leverage,  the  greater  must 
be  their  effect.  In  other  words,  all  the  material  resisting  compres- 
sion will  produce  its  greatest  effect  when  collected  the  farthest  possi- 
ble from  the  neutral  axis  at  the  top  of  the  beam  ;  and,  in  like  man- 
ner, all  the  material  resisting  extension  will  produce  its  greatest 
effect  when  similarly  disposed  at  the  bottom  of  the  beam.  We  are 
thus  directed  to  the  first  general  principle  of  the  distribution  of  the 
material  into  two  flanges — one  forming  the  top  and  the  other  the  bot- 
tom of  the  beam — joined  by  a  comparatively  slender  rib.  Associat- 
ing with  this  principle  the  relation  of  the  forces  of  extension  and 
compression  of  the  material  employed,  we  arrive  at  a  form  of  beam 
in  which  the  material  is  so  distributed,  that  at  the  instant  it  is  about 
to  break  by  extension  on  the  one  side,  it  is  about  to  break  by  com- 
pression on  the  other,  and  consequently  is  of  the 
strongest  form.  Thus,  supposing  that  it  is  re- 
quired to  determine  that  form  in  a  girder  of 
cast  iron:  the  ratio  of  the  crushing  force  of 
that  metal  to  the  force  of  extension  may  be 
taken  generally  as  6J  to  1,  which  is  therefore  also  the  ratio  of  the 
lower  to  the  upper  flange,  as  in  the  annexed  sectional  diagram. 

A  series  of  nine  castings  were  made,  gradually  increasing  tho 
lower  flange  at  the  expense  of  the  upper  one,  and  in  the  first  eight 


276 


THE  PRACTICAL  MODEL  CALCULATOR. 


experiments  the  beam  broke  by  the  tearing  asunder  of  the  lower 
flange ;  and  in  the  last  experiment  the  beam  yielded  by  the  crush- 
ing of  the  upper  flange.  In  the  eight  experiments  the  upper  flange 
was  therefore  the  weakest,  and  in  the  ninth  the  strongest,  so  that 
the  form  of  maximum  strength  was  intermediate,  and  very  closely 
allied  to  that  form  of  beam  employed  in  the  last  experiment,  which 
was  greatly  the  strongest.  The  circumstances  of  these  experiments 
are  contained  in  the  following  table. 


No.  of  experi- 
ments. 

Ratio  of  surfaces  of  com- 
pression and  extension. 

Area  of  cross  sections 
in  sq.  inches. 

Strength  per  sq.  inch 
of  sections  in  Ibs. 

1 

1  tol 

2-82 

2368 

2 

1  to  2 

2-87 

2567 

3 

Ito4 

3-02 

2737 

4 
5 

1  to4J 
Ito4 

3-37 
4-50 

3183 
3214 

6 

7 
8 

,      1  to  5J 
Ito3l 
1  to  4-3 

5-00 
4-628 

5-86 

3346 
3246 
3317 

9 

1  to  6-1 

6-4 

4075 

'    To  determine  the  weight  necessary  to  break  beams  cast  according 
to  the  form  described  : 

Multiply  the  area  of  the  section  of  the  lower  flange  by  the  depth 
of  the  beam,  and  divide  the  product  by  the  distance  between  the 
two  points  on  which  the  beam  is  supported :  this  quotient  multi- 
plied by  536  when  the  beams  are  cast  erect,  and  by  514  when  they 
are  cast  horizontally,  will  give  the  breaking  weight  in  cwts. 

From  this  it  is  not  to  be  inferred  that  the  beam  ought  to  have 
the  same  transverse  section  throughout  its  length.  On  the  con- 
trary, it  is  clear  that  the  section  ought  to  have  a  definite  relation 
to  the  leverage  at  which  the  load  acts.  From  a  mathematical  con- 
sideration of  the  conditions, 
it  indeed  appears  that  the 
effect  of  a  given  load  to 
break  the  beam  varies  when 
it  is  placed  over  different 
points  of  it,  as  the  products 
of  the  distances  of  these  points  from  the  points  of  support  of  the 
beam.  Thus  the  effect  of  a  weight  pkiced  at  the  point  W±  is  to  the 
effect  of  the  same  weight  acting  upon  the  point  W2,  as  the  product 
AWj  X  Wt  B  is  to  the  product  AW3  X  W3  B ;  the  points  of  sup- 
port being  at  A  and  B.  Since  then  the  effect  of  a  weight  increases 
as  it  approaches  the  middle  of  the  length  of  the  beam,  at  which  it 
is  a  maximum,  it  is  plain  that  the  beam  does  not  require  to  have 
the  same  transverse  section  near  to  its  extremities  as  in  the  middle  ; 
and,  guided  by  the  principle  stated,  it  is  easy  to  perceive  that  its 
strength  at  different  points  should  in  strictness  vary  as  the  products 
of  the  distances  of  these  points  from  the  points  of  support.  By 


STRENGTH    OF   MATERIALS.  277 

taking  this  law  as  a  fundamental  condition  in  the  distribution  of 
the  strength  of  a  beam,  whose  load  we  may  conceive  to  be  accumu- 
lated at  the  middle  of  its  length,  we  arrive  at  the  strongest  form 
which  can  be  attained  under  given  circumstances,  with  a  given 
amount  of  material ;  we  arrive  at  that  form  which  renders  the  beam 
equally  liable  to  rupture  at  every  point.  Now  this  form  of  maxi- 
mum strength  may  be  attained  in  two  ways;  either  by  varying  the 
depth  of  the  beam  according  to  the  law  stated,  or  by  preserving 
the  depth  everywhere  the  same,  and  varying  the  dimensions  of  the 
upper  and  lower  flanges  according  to  the  same  law.  The  conditions 
are  manifestly  identical.  We  may  therefore  assume  generally  the 
condition  that  the  section  is  rectangular,  and  that  the  thickness  of 
the  flanges  is  constant ;  then  the  outline  determined  by  the  law  in 
question,  in  the  one  case  of  the  elevation  of  the  beam  and  in  the 
other  of  the  plan  of  the  flanges,  is  the  geometrical  curve  called  a 
parabola — rather,  two  parabolas  joined  base  to  base  at  the  middle 
between  the  points  of  support.  The  annexed  diagram  represents 
the -plan  of  a  cast-iron  girder  according  to  this  form,  the  depth 


being  uniform  throughout.  Both  flanges  are  of  the  same  form, 
but  the  dimensions  of  the  upper  one  are  such  as  to  give  it  only  a 
sixth  of  the  strength  of  the  other. 

This,  it  will  be  observed,  is  also  the  form,  considered  as  an  ele- 
vation, of  the  beam  of  a  steam  engine,  which  good  taste  and  regard 
to  economy  of  material  have  rendered  common. 

It  must,  however,  be  borne  in  mind,  that  in  the  actual  practice 
of  construction,  materials  cannot  with  safety  be  subjected  to  forces 
approaching  to  those  which  produce  rupture.  In  machinery  espe- 
cially, they  are  liable  to  various  and  accidental  pressures,  besides 
those  of  a  permanent  kind,  for  which  allowance  must  be  made. 
The  engineer  must  therefore  in  his  practice  depend  much  on  expe- 
rience and  consideration  of  the  species  of  work  which  the  engine  is 
designed  to  perform.  If  the  engine  be  intended  for  spinning, 
pumping,  blowing,  or  other  regular  work,  the  material  may  be  sub- 
jected to  pressures  approaching  two-thirds  of  that  which  would  ac- 
tually produce  rupture ;  but  in  engines  employed  to  drive  bone- 
mills,  stampers,  breaking-down  rollers,  and  the  like,  double  that 
strength  will  often  be  found  insufficient.  In  cases  of  that  nature, 
experience  is  a  better  guide  than  theory. 

It  is  also  to  be  remarked  that  we  are  often  obliged  to  depart 
from  the  form  of  strength  which  the  calculation  gives,  on  account 
of  the  partial  strains  which  would  be  put  upon  some  of  the  parts 
of  a  casting,  in  consequence  of  unequal  cooling  of  the  metal  when 
the  thicknesses  are  unequal.  An  expert  founder  can  often  reduce 
the  irregular  contractions  which  thus  result;  but,  even  under  the 
best  management,  fracture  is  not  unfrequently  produced  by  irregu- 
Y 


278  THE   PRACTICAL   MODEL   CALCULATOR. 

larity  of  cooling,  ancKt  is  at  all  times  better  to  avoid  the  danger 
entirely,  than  to  endeavour  to  obviate  it  by  artifice.  For  this  rea- 
son, the  parts  of  a  casting  ought  to  be  as  nearly  as  possible  of  such 
thickness  as  to  cool  and  contract  regularly,  and  by  that  means  all 
partial  strain  of  the  parts  will  be  avoided. 

With  respect  to  design,  it  is  also  to  be  remarked,  that  mere  theo- 
retical properties  of  parts  will  not,  under  all  the  varieties  of  circum- 
stances which  arise  in  the  working  of  a  machine,  insure  that  exact 
adjustment  of  material  and  propriety  of  form  so  much  desired  in 
constructive  mechanics.  Every  design  ought  to  take  for  its  basis 
the  mathematical  conditions  involved,  and  it  would,  perhaps,  be  im- 
possible to  arrive  at  the  best  forms  and  proportions  by  any  more 
direct  mode  of  calculation ;  but  it  is  necessary  to  superadd  to  the 
mathematical  demonstration,  the  exercise  of  a  well-matured  judg- 
ment, to  secure  that  degree  of  adjustment  and  arrangement  of  parts 
in  which  the  merits  of  a  good  design  mainly  consist.  A  purely 
theoretical  engine  would  look  strangely  deficient  to  the  practised 
eye  of  the  engineer ;  and  the  merely  theoretical  contriver  would 
speedily  find  himself  lost,  should  he  venture  beyond  his  construction 
on  paper.  His  nice  calculations  of  the  "  work  to  be  performed,"  of 
the  vis  viva  of  the  mechanical  organs  of  his  machine,  and  of  the  modu- 
lus of  elasticity  of  his  material,  would,  in  practice,  alike  deceive  him. 

The  first  consideration  in  the  design  of  a  machine  is  the  quantity 
of  work  which  each  part  has  to  perform — in  other  words,  the  forces, 
active  and  inactive,  which  it  has  to  resist ;  the  direction  of  the 
forces  in  relation  to  the  cross-section  and  points  of  support ;  the 
velocity,  and  the  changes  of  velocity  to  which  the  moving  parts  are 
subject.  The  calculations  necessary  to  obtain  these  must  not  be 
confined  to  theory  alone ;  neither  should  they  be  entirely  deduced 
by  "rule  of  thumb;"  by  the  first  mode  the  strength  would,  in  all 
probability,  be  deficient  from  deficiency  of  material,  and  by  the 
second  rule  the  material  would  be  injudiciously  disposed ;  weight 
would  be  added  often  where  least  needed,  merely  from  the  deter- 
mination to  avoid  fracture,  and  in  consequence  of  a  want  of  know- 
ledge respecting  the  true  forms  best  adapted  to  give  strength. 

To  the  following  general  principles,  in  practice,  there  are  but 
few  real  exceptions : 

I.  Direct  Strain. — To  this  a  straight  line  must  be  opposed,  and 
if  the   part   be    of  considerable 

length,  vibration  ought  to  be  coun- 
teracted by  intersection  of  planes, 
(technically  feathers,}  as  repre- 
sented in  the  annexed  diagrams, 
or  $ome  such  form,  consistent  with  the  purpose  for  which  the  part 
is  intended. 

II.  Transverse  Strain. — To  this  a  parabolic  form  of  section  must 
be  opposed,  or  some  simple  figure  including  the  parabolic  form. 
For  economy  of  material,  the  vertex  of  the  curve  ought  to  be  at 
the  point  where  the  force  is  applied ;  and  when  the  strain  passes 


STRENGTH    OF   MATERIALS.  279 

alternately  from  one  side  of  the  part  to  the  other,  the  curve  ought 
to  be  on  both  sides,  as  in  the  beam  of  a  steam  engine. 

When  a  loaded  piece  is  supported  at  one  end  only,  if  the  breadth 
be  everywhere  the  same,  the  form  of  equal  strength  is  a  triangle ; 
but,  if  the  section  be  a  circle,  then  the  solid  will  be  that  generated 
by  the  revolution  of  a  semi-parabola  about  its  longer  axis.  In  prac- 
tice, it  will,  however,  be  sufficient  to  employ  the  frustum  of  a  cone, 
of  which,  in  the  case  of  cast  iron,  the  diameter  at  the  unsupported 
end  is  one-third  of  the  diameter  at  the  fixed  end. 

III.  Torsion. — The  section  most  commonly  opposed  to  torsion 
is  a  circle ;  and,  if  the  strain  be  applied  to  a  cylinder,  it  is  obvious 
the  rupture  must  first  take  place  at  the  surface,  where  the  torsion  is 
greatest,  and  that  the  further  the  material  is  placed  from  the  neutral 
axis,  the  greater  must  be  its  power  of  resistance ;  and  hence,  the 
amount  of  materials  being  the  same,  a  shaft  is  stronger  when  made 
hollow  than  if  it  were  made  solid. 

It  ought  not,  however,  to  be  supposed  that  the  circle  is  the  only 
figure  which  gives  an  axis  the  property  of  offering,  in  every  direc- 
tion, the  same  resistance  to  flexure.  On  the  contrary,  a  square  sec- 
tion gives  the  same  resistance  in  the  direction  of  its  sides,  and  of 
its  diagonals  ;  and,  indeed,  in  every  direction  the  resistance  is  equal. 
This  is,  moreover,  the  case  with  a  great  number  of  other  figures, 
which  may  be  formed  by  combining  the  circle  and  the  square  in  a 
symmetrical  manner ;  and  hence,  if  the  axis,  strengthened  by  salient 
sides,  as  in  feathered  shafts,  do  not  answer  as  well  as  cylindrical 
ones,  it  must  arise  from  their  not  being  so  well  disposed  to  resist 
torsion,  and  not  from  any  irregularities  of  flexure  about  the  axis 
inherent  in  the  particular  form  of  section. 

This  subject  has  been  investigated  with  much  care,  and,  accord- 
ing to  M.  Cauchy,  the  modulus  of  rupture  by  torsion,  T,  is  con- 
nected with  the  modulus  of  rupture  by  transverse  strain  S,  by  the 
simple  analogy  T  =  £  S. 

The  forms  of  all  the  parts  of  a  machine,  in  whatever  situation 
and  under  every  variety  of  circumstances,  may  be  deduced  from 
these  simple  figures ;  and,  if  the  calculations  of  their  dimensions 
be  correctly  determined,  the  parts  will  not  only  possess  the  requi- 
site degree  of  strength,  but  they  will  also  accord  with  the  general 
principles  of  good  taste. 

In  arranging  the  details  of  a  machine,  two  circumstances  ought 
to  be  taken  into  consideration.  The  first  is,  that  the  parts  subject 
to  wear  and  influenced  by  strain,  should  be  capable  of  adjustment ; 
the  second  is,  that  every  part  should,  in  relation  to  the  work  it  has 
to  perform,  be  equally  strong,  and  present  to  the  eye  a  figure  that 
is  consistent  with  its  degree  of  action.  Theory,  practice,  and  taste 
must  all  combine  to  produce  such  a  combination.  No  formal  law 
can  be  expressed,  either  by  words  or  figures,  by  which  a  certain 
contour  should  be  preferred  to  another ;  both  may  be  equally  strong 
and  equally  correct  in  reference  to  theory ;  custom,  then,  must  be 
appealed  to  as  the  guide. 


280 


THE    PRACTICAL    MODEL   CALCULATOR. 


TABLES  OP  THE  MECHANICAL  PROPERTIES  OF  THE  MATERIALS  MOST 
COMMONLY  EMPLOYED  IN  THE  CONSTRUCTION  OF  MACHINES  AiND 
FRAMINGS. 


NAMES. 

Specific 
Gravity. 

Weight  of 
cubic  ft. 
iu  Ibs. 

Tenacity  per 
square  inch 
in  Ibs. 

Crushing 
force  per  sq. 
in.  iu  Ibs 

Modulus  of 
elasticity 
in  Ibs. 

Mod.  of 
rupture 
nibs. 

Crushing 
force  to 
tenacity. 

TABLE  I.  —  Mechanical  Properties  of 
Ute  Common  Mttals. 

8*399 

525*00 

17968 

10304 

8930000 

0-573  '  1 

Cupper  (cast) 

8-607 

537-93 

549*06' 

19072 



5(H)*00 

61228 

ditto    (in  bolts)          
Iron  (English  wrought)  . 

7700 
f  7-760 

481-20 
475-50 

48000 
25%  tons 



24920000 

= 



i             tn 

\7-800 

487-00 

25%  tons 



,  .     ,         /  0             •                 '        V 

ditto  (Kussian)  in  wire,  l-20th  to  l-30th 

\ 

inch  diameter        .... 

j 

UltIO  ^r     let                       <                       0               ) 

ditto  in  chains,  oval  links,  6  inches  clear, 

\ 

iron  >2  inch  diameter  .        . 

) 

18014400 

44046 

' 

i  I*        /         ^^n^           •        •        •        • 

45828 

ditto      (Carron,  No.  2)  hot  Wast  . 
ditto      {     do.        do.  )  cold  blast     . 
ditto      (     do.      No.  3)  . 
ditto      (     do.        do.  >  hot  blast 

7-046 
7-060 
7-094 
7-056 

440-37 
441-62 
443-37 
441-00 

13505 
16683 
14200 
17755 

108540 
106375 
115442 
133440 

16085000 
17270500 
16246966 
17873100 

37503 
385.56 
33980 
42120 

3628  S 

8-037  :  1 
6-376  :  1 
8-129  :  1 
7-515  :  1 

7-229 

451-81 

21907 

145435 

22473650 

43497 

ditto      (Buft'rey.No,  1)  cold  blast  .      . 
ditto      (     do.        do.  }  hot  blast 
ditto      (Coed-Talon,  No.  2)  cold  blast 
ditto       (         do.             do.  )  hot  blast  . 
ditto      (         do           No  3)  cold  blast 

7-079 
6-998 
6-955 
6-968 
7-104 

442-43 
437-37 
434-06 
435-50 
449-62 

17466 
13434 

18855 
16676 

93366 
86397 
81770 
82739 

15381200 
13730500 
14313500 
14322500 
17102000 

37503 
35316 
33104 
33145 
43541 

5-346  :  1 
6-431  :  1 
4-a37  :  1 
4-961  :  1 

435-62 

14707(IOU 

40159 

ditto      (Milton  No  1)  hot  blast 

6-976 

436-00 

11974500 

•N',V> 

ditto      (Muirki'rk,  No.  1)  cold  blast   . 
ditto      (       do.          do.  )  hot  blast  . 
ditto      (Elsicar,  No.  1)  cold  blast  .      . 
Lead  (P:nglish  cast)         .... 
ditto  (milled-sheet) 

7-113 
6-953 
7-030 
11-446 

444-56 
434-56 
439-37 
717-45 
712-93 

1824 
3328 

:==L 

14003550 
13294400 
13981000 
720000 

86623 

;iv->n 

34862 

==: 

Silver  (standard) 



Mercury  (at  32°)             . 

851'18 

ditto      (at  60°) 

848'75 

ditto  (razor-tempered)       .... 
Tin  (cast)         
Zinc  (cast) 

7-840 

7-291 

490-00 
455-68 
439.95 

150000 
5322 



29000000 
4608000 
13680000 

— 



ditto  (rolled 

450-9 

TABLE  II.  —  Principal  Woods. 
Acacia  (English) 

44-37 

16000 

1152000 

11202 

Beech  {D™   '.'.'.-      '      • 

0-854 

53-37 
43-12 

15784 
17850 

7733      ( 
9363     \ 

13536000 

933(53 

0-55:1 

49-50 

15000 

6402 

1  r)62400 

10920 

Birch    |American      . 

40'50 

11633 

125761)0 

IChristiania  middle   . 
Merael  middle      . 

0-698 
0-5<X) 

43-62 

36-87 

12400 

1672000 
1535200 

9864 

lOSSS 



Norway  spruce 
English          .... 

0-310 

21-25 
29-37 

17600 
7000 





3675 

13489 

10331 

6WS40 

6078 

F.r       {New  England       .    _^   .        . 

0-553 
0-753 

34-56 

47*06 

12000 

6000 

2191200 

6612 

Larch  (seasoned)        

0-522 

32-62 

76-25 

10220 
11800 

5568 

1052800 

6894 

0-55:1 

Mahogany  (Spanish)  .... 

50*00 

16500 

8198 

f  English      

0*934 

58'37 

17300 

4684  wet  \ 

1451200 

10032 

f  0-28:1 

Oak     \  Canadian      .... 

54-50 

10253 

9504  dry/ 
4231  weft 

2148800 

10596 

1  0-57:1 
/  0-42:1 

\  Dantzio      
(Pitch      
Pine     4  Red     . 

0-756 
0-660 

47-24 
41-25 
41-06 

12780 
7818 

9509  dry  J 
5375 

1191200 
1225600 

1840000 

8748 
9792 
8146 

\  0-95:1 

(Yellow  

28'81 

5445 

1600000 

Poplar    .... 

3107  wet  1 

/  0-43:1 

5124  dry/ 
12101 

1  0-74:1 

Willow  (dry) 

Yew  (Spanish)  . 

'      ' 

STRENGTH   OF   MATERIALS. 


281 


THE   COHESIVE   STRENGTH   OF   BODIES. 

The  following  TABLE  contains  the  result  of  experiments  on  the 
cohesive  strength  of  various  bodies  in  avoirdupois  pounds  ;  also, 
one-third  of  the  ultimate  strength  of  each  body,  this  being  consi- 
dered sufficient,  in  most  cases,  for  a  permanent  load  : 


Names  of  Bodies. 

Square  Bar. 

tOne-third. 

Round  Bar. 

One-third. 

WOODS. 

Boxwood  

Ibs. 

20000 

Ibs. 

6667 

lb*. 

15708 

Ibs. 

5236 

Ash  

17000 

5667 

13357 

4452 

Teak 

15000 

5000 

11781 

3927 

Fir           

12000 

4000 

9424 

3141 

Beach  

11500 

3866 

9032 

3011 

Oak  

11000 

3667 

8639 

2880 

METALS. 

Cast  iron  

18656 

6219 

14652 

4884 

English  wrought  iron  
Swedish   do.   do  
Blistered  steel      ... 

55872 
72064 
133152 

18624 
24021 
44384 

43881 
56599 
104577 

14627 

18866 
34859 

Shear    do  

124400 

41366 

97703 

32568 

Cast     do  

134256 

44752 

105454 

35151 

Cast  copper  

19072 

6357 

14979 

4993 

Wrought  do 

33792 

11264 

26540 

8827 

Yellow  brass  .   . 

17968 

5989 

14112 

4704 

Cast  tin  

4736 

1579 

3719 

1239 

Cast  lead  

1824 

608 

1432 

477 

PROBLEM  I. 

.RULE. — To  find  the  ultimate  cohesive  strength  of  square,  round, 
and  rectangular  bars,  of  any  of  the  various  bodies,  as  specified  in 
the  table. — Multiply  the  strength  of  an  inch  bar,  (as  in  the  table,) 
of  the  body  required,  by  the  cross  sectional  area  of  square  and 
rectangular  bars,  or  by  the  square  of  the  diameter  of  round  bars ; 
and  the  product  will  be  the  ultimate  cohesive  strength. 

A  bar  of  cast  iron  being  1J  inches  square,  required  its  cohesive 
power. 

1-5  x  1-5  x  18656  =  41976  Ibs. 

Required  the  cohesive  force  of  a  bar  of  English  wrought  iron, 
2  inches  broad,  and  f  of  an  inch  in  thickness. 

2  x  -375  x  55872  =  41904  Ibs. 

Required  the  ultimate  cohesive  strength  of  a  round  bar  of 
wrought  copper  f  of  an  inch  in  diameter. 

•752  x  26540  =  14928-75  Ibs. 

PROBLEM    II. 

RULE. —  The  weight  of  a  body  being  given,  to  find  the  cross  sec- 
tional dimensions  of  a  bar  or  rod  capable  of  sustaining  that  weight. — 
For  square  and  round  bars,  divide  the  weight  given  by  one-third 
of  the  cohesive  strength  of  an  inch  bar,  (as«specified  in  the  table,) 
and  the  square  root  of  the  quotient  will  be  the  side  of  the  square, 
or  diameter  of  the  bar  in  inches. 

Y2 


282 


THE  PRACTICAL  MODEL  CALCULATOR. 


And  if  rectangular,  divide  the  quotient  by  the  breadth,  and  the 
result  will  be  the  thickness. 

What  must  be  the  side  of  a  square  bar  of  Swedish  iron  to  sus- 
tain a  permanent  weight  of  18000  Ibs  ? 


^ 


18000 


'  or  nearv"  8  °*  an  mc"  square. 


24021 

Required  the  diameter  of  a  round  rod  of  cast  copper  to  carry 
a  weight  of  6800  Ibs. 

6800 
V'TQQO  =  1.16  inches  diameter. 

A  bar  of  English  wrought  iron  is  to  be  applied  to  carry  a  weight 
of  2760  Ibs.  ;  required  the  thickness,  the  breadth  being  two  inches. 

=  '142  -f-  2  =  -071  of  an  inch  in  thickness. 


A  TABLE  showing  the  circumference  of  a  rope  equal  to  a  chain 
made  of  iron  of  a  given  diameter,  and  the  weight  in  tons  that 
each  is  proved  to  carry  ;  also,  the  iveight  of  a  foot  of  chain  made 
from  iron  of  that  dimension. 


Ropes. 
Cir.  in  Ins. 

Chains. 
Diam.  in  Inches. 

Proved  to  carry 
in  tons. 

Weight  of  a  lineal 
foot  in  Ibs.  Avr. 

3 

JandA 

1 

1-08 

4 

I 

2 

1-5 

4f 

fandA 

3 

2 

5* 

i 

4 

2-7 

6 

i  and  A 

5 

3-3 

6J 

f 

6 

4 

7 

8 

4-6 

7J 

| 

9f 

5-5 

8 

|  and  A 

6-1 

9 

I 

13 

7-2 

9* 

4  and  A 

15 

8-4 

10J 

1  inch. 

18 

9-4 

ON   THE   TRANSVERSE   STRENGTH   OF   BODIES. 

The  tranverse  strength  of  a  body  is  that  power  which  it  exerts 
in  opposing  any  force  acting  in  a  perpendicular  direction  to  its 
length,  as  in  the  case  of  beams,  levers,  &c.,  for  the  fundamental 
principles  of  which  observe  the  following: — 

That  the  transverse  strength  of  beams,  &c.  is  inversely  as  their 
lengths,  and  directly  as  their  breadths,  and  square  of  their  depths, 
and,  if  cylindrical,  as  the  cubes  of  their  diameters  ;  that  is,  if  a 
beam  6  feet  long,  2  inches  broad,  and  4  inches  deep,  can  carry 
2000  Ibs.,  another  beam  of  the  same  material,  12  feet  long,  2  inches 
broad,  and  4  inches  deep,  will  only  carry  1000,  being  inversely  as 
their  lengths.  Again,  if  a  beam  6  feet  long,  2  inches  broad,  and 
4  inches  deep,  can  support  a  weight  of  2000  Ibs.,  another  beam  of 


STRENGTH    OF    MATERIALS. 


283 


the  same  material,  6  feet  long,  4  inches  broad,  and  4  inches  deep, 
will  support  double  that  weight,  being  directly  as  their,  breadths ; 
— but  a  beam  of  that  material,  6  feet  long,  2  inches  broad,  and 
8  inches  deep,  will  sustain  a  weight  of  8000  Ibs. ;  being  as  the 
square  of  their  depths. 

From  a  mean  of  experiments  made,  to  ascertain  the  transverse 
strength  of  various  bodies,  it  appears  that  the  ultimate  strength 
of  an  inch  square,  and  an  inch  round  bar  of  each,  1  foot  long, 
loaded  in  the  middle,  and  lying  loose  at  both  ends,  is  nearly  as 
follows,  in  Ibs.  avoirdupois. 


Names  of  Bodies. 

Square  Bar. 

One-third. 

Round  Bar. 

One-third. 

Oak  

800 

267 

628 

209 

Ash  

1137 

379 

893 

298 

Elm 

569 

139 

447 

149 

Pitch  pine    .  .       .  .  •  • 

916 

305 

T19 

239 

Deal  

566 

188 

444 

148 

Cast  iron  

2580 

860 

2026 

675 

Wrought  iron 

4013 

1338 

3152 

1050 

PROBLEM   I. 

RULE.  —  To  find  the  ultimate  transverse  strength  of  any  rectan- 
gular beam,  supported  at  both  ends,  and  loaded  in  the  middle;  or 
supported  in  the  middle,  and  loaded  at  both  ends  ;  also,  when  the 
weight  is  between  the  middle  and  the  end  ;  likewise  when  fixed  at 
one  end  and  loaded  at  the  other.  —  Multiply  the  strength  of  an  inch 
square  bar,  1  foot  long,  (as  in  the  table,)  by  the  breadth,  and  square 
of  the  depth  in  inches,  and  divide  the  product  by  the  length  in 
feet  ;  the  quotient  will  be  the  weight  in  Ibs.  avoirdupois. 

What  weight  will  break  a  beam  of  oak  4  inches  broad,  8  inches 
deep,  and  20  feet  between  the  supports  ? 

800  x  4  x  82 

—  2Q—    -  =  10240  Ibs. 

When  a  beam  is  supported  in  the  middle,  and  loaded  at  each 
end,  it  will  bear  the  same  weight  as  when  supported  at  both  ends 
and  loaded  in  the  middle  ;  that  is,  each  end  will  bear  half  the 
weight. 

When  the  weight  is  not  situated  in  the  middle  of  the  beam,  but 
placed  somewhere  between  the  middle  and  the  end,  multiply  twice 
the  length  of  the  long  end  by  twice  the  length  of  the  short  end,  and 
divide  the  product  by  the  whole  length  of  the  beam  ;  the  quotient 
will  be  the  effectual  length. 

Required  the  ultimate  transverse  strength  of  a  pitch  pine  plank 
24  feet  long,  3  inches  broad,  7  inches  deep,  and  the  weight  placed 
8  feet  from  one  end. 


32  x  16 


=  21-3  effective  length. 


and 


916  x  3  x  7s 
21-3 


6321  Ibs. 


284  THE   PRACTICAL   MODEL   CALCULATOR. 

Again,  when  a  beam  is  fixed  at  one  end  and  loaded  at  the  other, 
it  will  only  bear  J  of  the  weight  as  when  supported  at  both  ends 
and  loaded  in  the  middle. 

What  is  the  weight  requisite  to  break  a  deal  beam  6  inches  broad, 
9  inches  deep,  and  projecting  12  feet  from  the  wall  ? 

566  X126X92  =  22923  -H4-  5730-7  Ibs. 

The  same  rules  apply  as  well  to  beams  of  a  cylindrical  form, 
with  this  exception,  that  the  strength  of  a  round  bar  (as  in  the 
table)  is  multiplied  by  the  cube  of  the  diameter,  in  place  of  the 
breadth,  and  square  of  the  depth. 

Required  the  ultimate  transverse  strength  of  a  solid  cylinder  of 
cast  iron  12  feet  long  and  5  inches  diameter. 

•  2026  X  53 

— j2 =  21104  Ibs. 

What  is  the  ultimate  transverse  strength  of  a  hollow  shaft  of 
cast  iron  12  feet  long,  8  inches  diameter  outside,  and  containing 
the  same  cross  sectional  area  as  a  solid  cylinder  5  inches  diameter  ? 


v/82  -  52  =  6-24,  and  83  -  6-243  =  269. 

Then,  ^^  =  45416  ,b, 

When  a  beam  is  fixed  at  both  ends,  and  loaded  in  the  middle,  it 
will  bear  one-half  more  than  it  will  when  loose  at  both  ends. 

And  if  a  beam  is  loose  at  both  ends,  and  the  weight  laid  uni- 
formly along  its  length,  it  will  bear  double;  but  if  fixed  at  both 
ends,  and  the  weight  laid  uniformly  along  its  length,  it  will  bear 
triple  the  weight. 

PROBLEM    II. 

RULE.  —  To  find  the  breadth  or  depth  of  beams  intended  to  sup- 
port a  permanent  weight.  —  Multiply  the  length  between  the  sup- 
ports, in  feet,  by  the  weight  to  be  supported  in  Ibs.,  and  divide  the 
product  by  one-third  of  the  ultimate  strength  of  an  inch  bar,  (as 
in  the  table,)  multiplied  by  the  square  of  the  depth  ;  the  quotient 
will  be  the  breadth,  or,  multiplied  by  the  breadth,  the  quotient  will 
be  the  square  of  the  depth,  both  in  inches. 

Required  the  breadth  of  a  cast  iron  beam  16  feet  long,  7  inches 
deep,  and  to  support  a  weight  of  4  tons  in  the  middle. 

8960  x  16 
4  tons  =  8960  Ibs.  and  —  3-4  inches. 


What  must  be  the  depth  of  a  cast  iron  beam  3-4  inches  broad, 
16  feet  long,  and  to  bear  a  permanent  weight  of  four  tons  in  the 
middle  ? 

8960  x  16 
^860x3-4  =  7  mches' 


STRENGTH   OF   MATERIALS.  285 

When  a  beam  is  fixed  at  both  ends,  the  divisor  must  be  multi- 
plied by  1*5,  on  account  of  it  being  capable  of  bearing  one-half 
more.  % 

When  a  beam  is  loaded  uniformly  throughout,  and  loose  at  both 
ends,  the  divisor  must  be  multiplied  by  2,  because  it  will  bear 
double  the  weight. 

If  a  beam  is  fast  at  both  ends,  and  loaded  uniformly  throughout, 
the  divisor  must  be  multipled  by  3,  on  account  that  it  will  bear 
triple  the  weight. 

Required  the  breadth  of  an  oak  beam  20  feet  long,  12  inches 
deep,  made  fast  at  both  ends,  and  to  be  capable  of  supporting  a 
weight  of  12  tons  in  the  middle. 

26880  x  20 
12  tons  =  26880  Ibs.,  and  266  x  122  x  1-5  ^  ^  *ncnes- 

Again,  when  a  beam  is  fixed  at  one  end,  and  loaded  at  the  other, 
the  divisor  must  be  multiplied  by  -25  ;  because  it  will  only  bear 
one-fourth  of  the  weight. 

Required  the  depth  of  a  beam  of  ash  6  inches  broad,  9  feet 
projecting  from  the  wall,  and  to  carry  a  weight  of  47  cwt. 

5264  x  9 
47  cwt.  =  5264  Ibs.,  and  ^379  x  6  x  -25  =  9'12  inches  deeP- 

And  when  the  weight  is  not  placed  in  the  middle  of  a  beam,  the 
effective  length  must  be  found  as  in  Problem  I. 

Required  the  depth  of  a  deal  beam  20  feet  long,  and  to  support 
a  weight  of  63  cwt.  6  feet  from  one  end. 

28  x  12 
—  2Q  —  ~  16-8  effective  length  of  beam,  and 

63  cwt.  =  7056  Ibs.  ;  hence 
,7056  x  16-8  , 

188  x  6  —  =  10*24  inches  deep. 

Beams  or  shafts  exposed  to  lateral  pressure  are  subject  to  all  the 
foregoing  rules,  but  in  the  case  of  water-wheel  shafts,  &c.,  some  al- 
lowances must  be  made  for  wear  ;  then  the  divisor  may  be  changed 
from  675  to  600  for  cast  iron. 

Required  the  diameter  of  bearings  for  a  water-wheel  shaft  12 
feet  long,  to  carry  a  weight  of  10  tons  in  the  middle. 
10  tons  =  22400  Ibs.,  and 

22400 

=  -\K448  =  7  '65  inches  diameter. 


And  when  the  weight  is  equally  distributed  along  its  length,  the 
cube  root  of  half  the  quotient  will  be  the  diameter,  thus  : 

448 

-g-  =  ^224  =  6-07  inches  diameter. 

Required  the  diameter  of  a  solid  cylinder  of  cast  iron,  for  the 
shaft  of  a  crane,  to  be  capable  of  sustaining  a  weight  of  10  tons  ; 


286 


THE  PRACTICAL  MODEL  CALCULATOR. 


one  end  of  the  shaft  to  be  made  fasf  in  the  ground,  the  other  to 
project  6  J  feet ;  and  the  effective  leverage  of  the  jib  as  If  to  1. 

10  .-tons  =  22400  Ibs.,  and 
22400  x  6-5  x  1-75 

675  x  -25  '    -  =  1509 


And  -^1509  =  11-47  inches  diameter. 

The  strength  of  cast  iron  to  wrought  iron,  in  this  direction,  is  as 
9  is  to  14  nearly ;  hence,  if  wrought  iron  is  taken  in  place  of  cast 
iron  in  the  last  example,  what  must  be  its  diameter  ? 

1509  x  9 
*ty TJ; — •  =  9-89  inches  diameter. 

ON   TORSION   OR   TWISTING. 

The  strength  of  bodies  to  resist  torsion,  or  wrenching  asunder, 
is  directly  as  the  cubes  of  their  diameters ;  or,  if  square,  as  the 
cube  of  one  side  ;*  and  inversely  as  the  force  applied  multiplied  into 
the  length  of  the  lever. 

Hence  the  rule. — 1.  Multiply  the  strength  of  an  inch  bar,  by 
experiment,  (as  in  the  following  table,)  by  the  cube  of  the  diameter, 
or  of  one  side  in  inches ;  and  divide  by  the  radius  of  the  wheel,  or 
length  of  the  lever  also  in  inches ;  and  the  quotient  will  be  the  ul- 
timate strength  of  the  shaft  or  bar,  in  Ibs.  avoirdupois. 

2. — Multiply  the  force  applied  in  pounds  by  the  length  of  the 
lever  in  inches,  and  divide  the  product  by  one-third  of  the  ultimate 
strength  of  an  inch  bar,  (as  in  the  table,)  and  the  cube  root  of  the 
quotient  will  be  the  diameter,  or  side  of  a  square  bar  in  inches ; 
that  is,  capable  of  resisting  that  force  permanently. 

The  following  TABLE  contains  the  result  of  experiments  on  inch  bars, 
of  various  metals,  in  Ibs.  avoirdupois. 


Names  of  Bodies. 

Round  Bar. 

One-third. 

Square  Bar. 

One-third. 

Cast  iron  

11943 

3981 

15206 

5069 

English  wrought  iron 
Swedish      do.      do. 
Blistered  steel  

12063 
11400 
20025 

4021 
3800 
6675 

15360 
14592 
25497 

5120 

4864 
8499 

Shear  do  
Cast  do  

20508 
21111 
5549 

6836 
7037 
1850 

26112 
26880 
7065 

8704 
8960 
2355 

Cast  copper         

4825 

1608 

6144 

2048 

Tin  

1688 

563 

2150 

717 

Lead  

1206 

402 

1536 

512 

What  weight,  applied  on  the  end  of  a  5  feet  lever,  will  wrench 
asunder  a  3  inch  round  bar  of  cast  iron  ? 

11943  x  33       ,       •„  .,      . 

— QQ =  5374  Ibs.  avoirdupois. 

Required  the  side  of  a  square  bar  of  wrought  iron,  capable  of  re- 
sisting the  twist  of  600  Ibs.  on  the  end  of  a  lever  8  feet  long. 
600  x  96 


STRENGTH    OF   MATER 


287 


In  the  case  of  revolving  shafts  for  machinery,  &c.,  the  strength 
is  directly  as  the  cubes  of  their  diameters,  and  revolutions,  and  in- 
versely as  the  resistance  they  have  to  overcome  ;  hence, 

From  practice,  we  find  that  a  40  horse  power  steam  engine, 
making  25  revolutions  per  minute,  requires  a  shaft  (if  made  of 
wrought-iron)  to  be  8  inches  diameter  :  now,  the  cube  of  8,  multi- 
plied by  25,  and  divided  by  40  =  320  ;  which  serves  as  a  constant 
multiplier  for  all  others  in  the  same  proportion. 

What  must  be  the  diameter  of  a  wrought  iron  shaft  for  an  engine 
of  65  horse  power,  making  23  revolutions  per  minute  ? 


65  X  320 
--  03  - 


=  9*67  inches  diameter. 


James  Glenie,  the  mathematician,  gives  400  as  a  constant  mul- 
tiplier for  cast  iron  shafts  that  are  intended  for  first  movers  in  ma- 
chinery ; 

200  for  second  movers  ;  and 
100  for  shafts  connecting  smaller  machinery,  &c. 
The  velocity  of  a  30  horse  power  steam  engine  is  intended  to  be 
19  revolutions  per  minute.     Required  the  diameter  of  bearings  for 
the  fly-wheel  shaft. 


,400  x  30 
19 


8-579  inches  diameter. 


Required  the  diameter  of  the  bearings  of  shafts,  as  second  movers 
from  a  30  horse  engine  ;  their  velocity  being  36  revolutions  per 
minute. 


200  x  30 


=  5*5  inches  diameter. 


When  shafting  is  intended  to  be  of  wrought  iron,  use  160  as  the 
multiplier  for  second  movers  ;  and  80  for  shafts  connecting  smaller 
machinery. 

TABLE  of  the  Proportionate  Length  of  Bearings,  or  Journals  for 
Shafts  of  vdrious  diameters. 


Dia.  in  Inches. 

Len.  in  Inches. 

Dia.  in  Inches. 

Len.  in  Inches. 

1 

1* 

6i 

8f 

1| 

j      7 

9f 

2 

3 

71 

10 

2i 

§t 

8 

10f 

5% 

3J 

8i 

lit 

3 

4J 

9" 

12 

3J 

4|   ' 

Ql 
2 

12f 

4 

10 

13J 

4} 

6| 

10^ 

14 

5 

6| 

11 

14J 

5J 

HJ 

15J 

6 

8i 

12 

16 

288 


THE  PRACTICAL  MODEL  CALCULATOR. 


Tenacities,  Resistances  to  Compression,  and  other  Properties  of  the 
common  Materials  of  Construction. 


Names  of  Bodies. 

Absolute. 

Compared  with  Cast  Iron. 

Tenacity  in  Ibs. 
per  sq.  inch. 

Resistance  to 
compression 
in  Ibs.  per  sq. 
in. 

Its  strength 
ia 

Its  extensi- 
bility is 

Its  stiffness  is 

Ash  

14130 
12225 
17968 
275 
13434 
33000 
9720 
12346 
11800 
11835 

0-23 
0-15 
0-435 

1-000, 

0-21 
0-23 
0-3 
0-25 

0-6& 
1-12 
0-136 
0-096 
0-24 

0-25 

0-182 
0-365 

2-6 
2-1 
0-9 

1-0 

2-9 
2-4 
2-4 

2-9 

1-25 
0-86 
2-3 
2-5 
2-9 

2-8 

0-75 
0-5 

0-089 
0-073 
0-49 

1-000 

0-073 
0-1 
0-1 

0-087 

0-535 
1-3 
0-058 
0-0385 

0-487 

0-093 

0-25 
0-76 

Beech 

8548 
10304 
562 
86397 

1033 
2028 
5375 
5445 
10910 

Brass    . 

Brick  

Copper  (wrought)  
Elm   

Fir,  or  Pine,  white  
red 

—     yellow.... 

Gun-metal    (copper   8, 
and  tin  1) 

35838 
56000 
12240 
1824 
11475 
551 
11880 
200 
128000 
478 
772 
2661 
857 
4736 
9120 

Malleable  iron    

Larch  

5568 

8000 
6060 
9504 

5490 
6630 
3729 

Lead  

Mahogany,  Honduras.. 
Marble  

Oak  

Rope  (1  in.  in  circum.) 
Steel               

Stone   Bath  

Craigleith 

Dundee       .  . 

Portland  

Tin  (cast)  

Zinc  (sheet)       

Comparative  Strength  and  Weight  of  Ropes  and  Chains. 


3.2 


2f 
4f 
51 

7 

111 
15 
19 
21 


-2.2 

If 


H 


s-s 


8 

10| 
14 
18 
22 
27 
32 
37 


Proof 
strength  in 
tons  &  cwt. 


1  16| 

2  10 

3  5i 

4  3} 

5  2 

6  4| 

7  7 

8  13i 


10 

lOf 

Hi 

1? 

13f 


23 

28 

36^ 

39 

45 


56 
16^     60 


1-2 

8.S 


lin. 

it 


43 
49 
56 
63 
71 
79 
87 
96 
106 


Proof  strength 
n  tons  &  cwt. 


10  0 

11  11 

13  8 

14  18 
16  14 

18  11 
20  8 
22  13 
24  18 


It  must  be  understood  and  also  borne  in  mind,  that  in  estimating 
the  amount  of  tensile  strain  to  which  a  body  is  subjected,  the  weight 
of  the  body  itself  must  also  be  taken  into  account ;  for  according 
to  its  position  so  may  it  approximate  to  its  whole  weight,  in  tend- 


STRENGTH    OF    MATERIALS. 


289 


ing  to  produce  tension  within  itself;  as  in  the  almost  constant 
application  of  ropes  and  chains  to  great  depths,  considerable 
heights,  &c. 

Alloys  that  are  of  greater  Tenacity  than  the  sum  of  their  Constitu- 
ents, as  determined  by  the  Experiments  of  Muschenbroek. 

Swedish  copper  6  parts,  Malacca  tin  1 — tenacity  per  square  inch  64,000  Ibs. 

Chili  copper  6  parts,  Malacca  tin  1 60,000 

Japan  copper  5  parts,  Banca  tin  1 57,000 

Anglesea  copper  6  parts,  Cornish  tin  1 41,000. 

Common  block  tin  4,  lead  1,  zinc  1 13,000 

Malacca  tin  4,  regulus  of  antimony  1 12,000 

Block  tin  3,  lead  1 10,200 

Block  tin  8,  zinc  1 10,000 

Lead  1,  zinc  1 4,500 

TABLE  of  Data,  containing  the  Results  of  Experiments  on  the  Elas- 
ticity and  Strength  of  various  Species  of  Timber. 


Species  of  Timber. 

Value  of  E. 

Value  of  S. 

Species  of  Timber. 

Value  of  E. 

Value  of  S. 

Teak 

174-7 

2462 

Elm  

50-64 

1013 

Poona 

122-26 

2221 

Pitch  pine 

88-68 

1632 

English  oak  . 

105 

1672 

Red  pine 

133 

1341 

Canadian  do  
Dantzic     do  

155-5 

86-2 

1766 
1457 

New  England  fir 
Riga  fir  

158-5 
90 

1102 
1100 

Adriatic    do  
Ash  

70-5 
119 

1383 
2026 

Mar  Forest     do. 

63 
76 

1200 
900 

Beech 

98 

1556 

Norway  spruce 

105-47 

1474 

RULE. — To  find  the  dimensions  of  a  beam  capable  of  sustaining 
a  given  weight,  with  a  given  degree  of  deflection,  when  supported 
at  both  ends. — Multiply  the  weight  to  be  supported  in  Ibs.  by  the 
cube  of  the  length  in  feet ;  divide  the  product  by  32  times  the 
tabular  value  of  E,  multiplied  into  the  given  deflection  in  inches, 
and  the  quotient  is  the  breadth  multiplied  by  the  cube  of  the  depth 
in  inches. 

When  the  beam  is  intended  to  be  square,  then  the  fourth  root 
of  the  quotient  is  the  breadth  and  depth  required. 

If  the  beam  is  to  be  cylindrical,  multiply  the  quotient  by  1'7, 
and  the  fourth  root  of  the  product  is  the  diameter. 

The  distance  between  t'he  supports  of  a  beam  of  Riga  fir  is 
16  feet,  and  the  weight  it  must  be  capable  of  sustaining  in  the 
middle  of  its  length  is  8000  Ibs.,  with  a  deflection  of  not  more 
than  |  of  an  inch ;  what  must  be  the  depth  of  the  beam,  suppos- 
ing the  breadth  8  inches  ? 

16  X  8000 
90  x  32  x  -75  ^  151T5  "*"  8  =  ^189T  ^  12'35  in>  tlie  dePth- 

RULE. — To  determine  the  absolute  strength  of  a  rectangular  beam 
of  timber  when  supported  at  both  ends,  and  loaded  in  the  middle 
of  its  length,  as  beams  in  general  ought  to  be  calculated  to,  so  that 
they  may  be  rendered  capable  of  withstanding  all  accidental  cases 
of  emergency. — Multiply  the  tabular  value  of  S  by  four  times  the 
depth  of  the  beam  in  inches,  and  by  the  area  of  the  cross  section 
in  inches  ;  divide  the  product. by  the  distance  between  the  supports 
Z  19 


290  THE   PRACTICAL   MODEL   CALCULATOR. 

in  inches,  and  the  quotient  will  be  the  absolute  strength  of  the 
beam  in  Ibs. 

If  the  beam  be  not  laid  horizontally,  the  distance  between  the 
supports,  for  calculation,  must  be  the  horizontal  distance. 

One-fourth  of  the  weight  obtained  by  the  rule  is  the  greatest 
weight  that  ought  to  be  applied  in  practice  as  permanent  load. 

If  the  load  is  to  be  applied  at  any  other  point  than  the  middle, 
then  the  strength  will  be,  as  the  product  of  the  two  distances  is  to 
the  square  of  half  the  length  of  the  beam  between  the  supports  ; 
or,  twice  the  distance  from  one  end,  multiplied  by  twice  from  the 
other,  and  divided  by  the  whole  length,  equal  the  effective  length 
of  the  beam. 

In  a  building  18  feet  in  width,  an  engine  boiler  of  5J  tons  is  to 
be  fixed,  the  centre  of  which  to  be  7  feet  from  the  wall  ;  and  having 
two  pieces  of  red  pine  10  inches  by  6,  which  I  can  lay  across  the 
two  walls  for  the  purpose  of  slinging  it  at  each  end,  —  may  I  with 
sufficient  confidence  apply  them,  so  as  to  effect  this  object  ? 

2240  x  5-5 

—  -  --  =  6160  Ibs.  to  carry  at  each  end. 

14  x  22 
And  18  feet  —  7  =  11,  double  each,  or  14  and  22,  then       lg 

=  17  feet,  or  204  inches,  effective  length  of  beam. 

1341  x  4  x  10  x  60 
Tabular  value  of  S,  red  pine  =  -          —  ^94  —  "  ~  *««  •* 

Ibs.,  the  absolute  strength  of  each  piece  of  timber  at  that  point. 

RULE.  —  To  determine  the  dimensions  of  a  rectangular  beam  capa- 
ble of  supporting  a  required  w  eight  ,  with  a  given  degree  of  deflection, 
when  fixed  at  one  end.  —  Divide  the  weight  to  be  supported,  in  Ibs., 
by  the  tabular  value  of  E,  multiplied  by  the  breadth  and  deflection, 
both  in  inches  ;  and  the  cube  root  of  the  quotient,  multiplied  by 
the  length  in  feet,  equal  the  depth  .required  in  inches. 

A  beam  of  ash  is  intended  to  bear  a  load  of  700  Ibs.  at  its  ex- 
tremity ;  its  length  being  5  feet,  its  breadth  4  inches,  and  the  de- 
flection not  to  exceed  J  an  inch. 

Tabular  value  of  E  =  119  X  4  X  -5  =  238,  the  divisor  ;  then 
700  -*-  238  =  ^/2-94  x  5  =  7'25  inches,  depth  of  the  beam. 

RULE.  —  To  find  the  absolute  strength  of  a  rectangular  beam,  ivhen 
fixed  at  one  end,  and  loaded  at  the  other.  —  Multiply  the  value  of  S 
by  the  depth  of  the  beam,  and  by  the  area  of  its  section,  both  in 
inches  ;  divide  the  product  by  the  leverage  in  inches,  and  the  quo- 
tient equal  the  absolute  strength  of  the  beam  in  Ibs. 

A  beam  of  Riga  fir,  12  inches  by  4J,  and  projecting  6J  feet  from 
the  wall  ;  what  is  the  greatest  weight  it  will  support  at  the  ex- 
tremity of  its  length  ? 

Tabular  value  of  S  =  1100 
12  X  4-5  =  54  sectional  area, 


Then,  =  9138-4  Ibs. 


STRENGTH   OF   MATERIALS.  291 

When  fracture  of  a  beam  is  produced  by  vertical  pressure,  the 
fibres  of  the  lower  section  of  fracture  are  separated  by  extension, 
whilst  at  the  same  time  those  of  the  upper  portion  are  destroyed 
by  compression ;  hence  exists  a  point  in  section  where  neither  the 
one  nor  the  other  takes  place,  and  which  is  distinguished  as  the 
point  of  neutral  axis.     Therefore,  by  the  law  of  fracture  thus  esta- 
blished, and  proper  data  of  tenacity  and  compression  given,  as  in 
the  Table  (p.  281),  we  are  enabled  to  form  metal  beams  of  strongest 
section  with  the  least  possible  material :  thus,  in  cast  iron  the  re- 
sistance to  compression  is  nearly  as  6J  to  1  of  tenacity ;  conse- 
quently a  beam  of  cast  iron,   to  be  of  strongest  section,  must  be 
of  the  form  TB,  and  a  parabola  in  the  direction  of"  its         c^p 
length,  the  quantity  of  material  in  the  bottom  flange 
being  about  6  J  times  that  of  the  upper :  but  such  is  not          u 
the  case  with  beams  of  timber ;  for  although  the  tenacity 
of  timber  be  on  an  average  twice  that  of  its  resistance  to  compres- 
sion, its  flexibility  is  so  great,  that  any  considerable  length  of  beam, 
where  columns  cannot   be  situated  to  its  support,  requires  to  be 
strengthened  or  trussed  by  iron  rods,  as  in  the  following  manner : 


And  these  applications  of  principle  not  only  tend  to  dimmish  de- 
flection, but  the  required  purpose  is  also  more  effectively  attained, 
and  that  by  lighter  pieces  of  timber. 

RULE. —  To  ascertain  the  absolute  strength  of  a  cast  iron  beam  of 
the  preceding  form,  or  that  of  strongest  section. — Multiply  the  sec- 
tional area  of  the  bottom  flange  in  inches  by  the  depth  of  the  beam 
in  inches,  and  divide  the  product  by  the  distance  between  the  sup- 
ports also  in  inches  ;  and  514  times  the  quotient  equal  the  absolute 
strength  of  the  beam  in  cwts. 

The  strongest  form  in  which  any  given  quantity  of  matter  can 
be  disposed  is  that  of  a  hollow  cylinder ;  and  it  has  been  demon- 
strated that  the  maximum  of  strength  is  obtained  in  cast  iron,  when 
the  thickness  of  the  annulus  or  ring  amounts  to  |th  of  the  cylinder's 
external  diameter ;  the  relative  strength  of  a  solid  to  that  of  a 
hollow  cylinder  being  as  the  diameters  of  their  sections. 

The  following  table  shows  the  greatest  weight  that  ever  ought 
to  be  laid  upon  a  beam  for  permanent  load,  and  if  there  be  any 
liability  to  jerks,  &c.,  ample  allowance  must  be  made;  also,  the 
weight  of  the  beam  itself  must  be  included. 

RULE. — To  find  the  weight  of  a  cast  iron  beam  of  given  dimen- 
sions.— Multiply  the  sectional  area  in  inches  by  the  length  in  feet, 
and  by  3*2,  the  product  equal  the  weight  in  Ibs. 

Required  the  weight  of  a  uniform  rectangular  beam  of  cast  iron, 
16  feet  in  length,  11  inches  in  breadth,  and  1J  inch  in  thickness. 
11  x  1-5  x  16  x  3-2  =  844-8  Ibs. 


292 


THE  PRACTICAL  MODEL  CALCULATOR. 


A  TABLE  showing  the  Weight  or  Pressure  a  Beam  of  Oast  Iron, 
1  inch  in  breadth,  will  sustain  without  destroying  its  elastic  force, 
when  it  is  supported  at  each  end,  and  loaded  in  the  middle  of  its 
length,  and  also  the  deflection  in  the  middle  which  that  weight 
ivill  produce. 


length. 

6  feet. 

7  feet. 

8  feet. 

9  feet. 

10  feet. 

Depth 

Wt.  in 

Defl.  in 

Wt.  in 

Defl.  in 

Wt.  in 

Defl.  in 

Wt.  in 

Defl.  in 

Wt.  in 

Defl.  in 

in  in. 

Ibs. 

in. 

Ibs. 

in. 

Ibs. 

in. 

Ibs. 

in. 

Ibs. 

in. 

3 

1278 

•24 

1089 

•33 

954 

•426 

855 

•54 

765 

•66 

3£ 

1739 

•205 

1482 

•28 

1298 

•365 

1164 

•46 

1041 

•57 

4 

2272 

•18 

1936 

•245 

1700 

•32 

1520 

•405 

1360 

•5 

4£ 

2875 

•16 

2450 

•217 

2146 

•284 

1924 

•36 

1721 

•443 

5 

3560 

•144 

3050 

•196 

2650 

•256 

2375 

•32 

2125 

•4 

6 

5112 

•12 

4356 

•163 

3816 

•213 

3420 

•27 

3060 

•33 

7 

6958 

•103 

5929 

•14 

5194 

•183 

4655 

•23 

4165 

•29 

8 

9088 

•09 

7744 

•123 

6784 

•16 

6080 

•203 

5440 

•25 

9 





9801 

•109 

8586 

•142 

7695 

•18 

6885 

•22 

10 



— 

12100 

•098 

10600 

•128 

9500 

•162 

8500 

.2 

11 



— 

— 

— 

12826 

•117 

11495 

•15 

10285 

•18 

12 







— 

15264 

•107 

13680 

•135 

12240 

•17 

13 





— 

— 

— 

— 

16100 

•125 

14400 

•154 

14 

— 

— 

— 

— 

— 

— 

18600 

•115 

16700 

•143 

12  feet. 

14  feet. 

16  feet. 

18  feet. 

20  feet. 

6 

2548 

•48 

2184 

•65 

19^2 

•85 

1699 

1-08 

1530 

1-34 

7 

3471 

•41 

2975 

•58 

2603 

•73 

2314 

•93 

2082 

1-14 

8 

4532 

•36 

3884 

»49 

3396 

•64 

3020 

•81 

2720 

1-00 

9 

5733 

'32^ 

4914 

•44 

4302 

•57 

3825 

•72 

3438 

•89 

10 

7083 

•28  ' 

6071 

•39 

5312 

•51 

4722 

•64 

4250 

•8 

11 

8570 

•26 

7346 

•36 

6428 

•47 

5714 

•59 

5142 

•73 

12 

10192 

•24 

8736 

•33 

7648 

•43 

6796 

•54 

6120 

•67 

13 

11971 

•22 

10260 

•31 

8978 

•39 

7980 

•49 

7182 

•61 

14 

13883 

•21 

11900 

•28 

10412 

•36 

9255 

•46 

8330 

•57 

15 

15937 

•19 

13660 

•26 

11952 

•34 

10624 

•43 

9562 

•53 

16 

18128 

•18 

15536 

•24 

13584 

•32 

12080 

•40 

10880 

•5 

17 

20500 

•17 

17500 

•23 

15353 

•3 

13647 

•38 

12282 

•47 

18 

22932 

•18 

19656 

•21 

17208 

•28 

15700 

•36 

13752 

•44 

Resistance  of  Bodies  to  Flexure  by  Vertical  Pressure. — When  a 
piece  of  timber  is  employed  as  a  column  or  support,  its  tendency 
to  yielding  by  compression  is  different  according  to  the  proportion 
between  its  length  and  area  of  its  cross  section ;  and  supposing  the 
form  that  of  a  cylinder  whose  length  is  less  than  seven  or  eight  times 
its  diameter,  it  is  impossible  to  bend  it  by  any  force  applied  longi- 
tudinally, as  it  will  be  destroyed  by  splitting  before  that  bending 
can  take  place ;  but  when  the  length  exceeds  this,  the  column  will 
bend  under  a  certain  load,  and  be  ultimately  destroyed  by  a  similar 
kind  of  action  to  that  which  has  place  in  the  transverse  strain. 

Columns  of  cast  iron  and  of  other  bodies  are  also  similarly  cir- 
cumstanced. 

When  the  length  of  a  cast  iron  column  with  flat  ends  equals 
about  thirty  times  its  diameter,  fracture  will  be  produced  wholly  by 
bending  of  the  material ; — when  of  less  length,  fracture  takes  place 
partly  by  crushing  and  partly  by  bending :  but,  when  the  column 


STRENGTH    OP   MATERIALS. 


293 


is  enlarged  in  the  middle  of  its  length  from  one  and  a  half  to  twice 
its  diameter  at  the  ends,  by  being  cast  hollow,  the  strength  is 
greater  by  {th  than  in  a  solid  column  containing  the  same  quantity 
of  material. 

RULE. — To  determine  the  dimensions  of  a  support  or  column  to 
bear  without  sensible  curvature  a  given  pressure  in  the  direction  of 
its  axis. — Multiply  the  pressure  to  be  supported  in  Ibs.  by  the 
square  of  the  column's  length  in  feet,  and  divide  the  product  by 
twenty  times  the  tabular  value  of  E  ;  and  the  quotient  will  be  equal 
to  the  breadth  multiplied  by  the  cube  of  the  least  thickness,  both 
being  expressed  in  inches. 

When  the  pillar  or  support  is  a  square,  its  side  will  be  the  fourth 
root  of  the  quotient. 

If  the  pillar  or  column  be  a  cylinder,  multiply  the  tabular 
value  of  E  by  12,  and  the  fourth  root  of  the  quotient  equal  the 
diameter. 

What  should  be  the  least  dimensions  of  an  oak  support,  to  bear 
a  weight  of  2240  Ibs.  without  sensible  flexure,  its  breadth  being  3 
inches,  and  its  length  5  feet  ? 

2240  x  53  

Tabular  value  of  E  =  105,  and  2Q  x  105  x  3  =  ^8'888  = 

2-05  inches. 

Required  the  side  of  a  square  piece  of  Riga  fir,  9  feet  in  length, 
to  bear  a  permanent  weight  of  6000  Ibs. 

6000  X  92 
Tabular  value  of  E  =  96,  and    9n  v  Qft    =  V253  =  4  inches 

i  tu\J    ?\    i/O 

nearly. 

Dimensions  of  Cylindrical  Columns  of  Cast  Iron  to  sustain  a  given 
load  or  pressure  with  safety. 


Diameter  in 
inches. 

Length  or  hajght  in  feet. 

4  |  6 

8  |  10 

12  |  14 

16 

18 

20  |  22  |  24 

Weight  or  load  in  cwts. 

2 

72 

60 

49 

40 

32 

26 

22- 

18 

15 

13 

11 

2J 

119 

105 

91 

77 

65 

55 

47 

40 

34 

29 

25 

3 

178 

163 

145 

128 

111 

97 

84 

73 

64 

56 

49 

3£ 

247 

232 

214 

191 

172 

156 

135 

119 

106 

94 

83 

4 

326 

310 

288 

266 

242 

220 

198 

178 

160 

144 

130 

*z 

418 

400 

379 

354 

327 

301 

275 

251 

229 

208 

189 

5 

522 

501 

479 

452 

427 

394 

365 

337 

310 

285 

262 

6 

607 

592 

573 

550 

525 

497 

469 

440 

413 

386 

360 

7 

1032 

1013 

989 

959 

924 

887 

848 

808 

765 

725 

686 

8 

1333 

1315 

1289 

1259 

1224 

1185 

1142 

1097 

1052 

1005 

959 

9 

1716 

1697 

1672 

1640 

1603 

1561 

1515 

1467 

1416 

1364 

1311 

10 

2119 

2100 

2077 

2045 

2007 

1964 

1916 

1865 

1811 

1755 

1697 

11 

2570 

2550 

2520 

2490 

2450 

2410 

2358 

2305 

2248 

2189 

2127 

12 

3050 

3040 

3020 

2970 

2930 

2900 

2830 

2780 

2730|  2670 

2600 

Practical  utility  of  the  preceding  Table. — Wanting  to  support  the 
front  of  a  building  with  cast  iron  columns  18  feet  in  length,  8  inches 
in  diameter,  and  the  metal  1  inch  in  thickness ;  what  weight  may 


294  THE    PRACTICAL   MODEL   CALCULATOR. 

I  confidently  expect  each  column  capable  of  supporting  without 
tendency  to  deflection  ? 

'    Opposite  8  inches  diameter  and  under  18  feet  =  1097 
Also  opposite  6  in.  diameter  and  under  18  feet  =    440 

=    65T  cwts. 

The  strength  of  cast  iron  as  a  column  being  =  1-0000 
steel  =  2-518 

wrought  iron       —  =  1-745 

oak  (Dantzic)    , —  =    -1088 

red  deal  =    -0785 

Elasticity  of  torsion,  or  resistance  of  bodies  to  twisting. — The 
angle  of  flexure  by  torsion  is  as  the  length  and  extensibility  of  the 
body  directly,  and  inversely  as  the  diameter ;  hence,  the  length 
of  a  bar  or  shaft  being  given,  the  power,  and  the  leverage  the 
power  acts  with,  being  known,  and  also  the  number  of  degrees  of 
torsion  that  will  not  affect  the  action  of  the  machine,  to  determine 
the  diameter  in  cast  iron  with  a  given  angle  of  flexure. 

RULE. — Multiply  the  power  in  Ibs.  by  the  length  of  the  shaft  in 
feet,  and  by  the  leverage  in  feet ;  divide  the  product  by  fifty-five 
times  the  number  of  degrees  in  the  angle  of  torsion,  and  the  fourth 
root  of  the  quotient  equal  the  shaft's  diameter  in  inches. 

Required  the  diameters  for  a  series  of  shafts  35  feet  in  length, 
and  to  transmit  a  power  equal  to  1245  Ibs.,  acting  at  the  circum- 
ference of  a  wheel  2J  feet  radius,  so  that  the  twist  of  the  shafts 
on  the  application  of  the  power  may  not  exceed  one  degree. 

1245  x  35  x  2-5 

55  x  i =  4v/l981  =  6-67  inches  in  diameter. 

Relative  strength  of  metals  to  resist  torsion. 

Cast  iron =  1          Swedish  bar  iron  ...=  1-05 

Copper =    -48    English       do =  1-12 

Yellow  brass =    -511  Shear  steel =  1-96 

Gun-metal =    -55     Cast     do =  2-1 

DEFLEXION  OF  RECTANGULAR  BEAMS. 

RULE. — To  ascertain  the  amount  of  deflexion  of  a  uniform  beam 
of  cast  iron,  supported  at  both  ends,  and  loaded  in  the  middle  to  the 
extent  of  its  elastic  force. — Multiply  the  square  of  the  length  in  feet 
by  -02,  and  the  product  divided  by  the  depth  in  inches  equal  the 
deflexion. 

Required  the  deflection  of  a  cast  iron  beam  18  feet  long  between 
the  supports,  12*8  inches  deep,  2-56  inches  in  breadth,  and  bear- 
ing a  weight  of  20,000  Ibs.  in  the  middle  of  its  length. 

182  x  *02 

— jHTg —  =  -506  inches  from  a  straight  line  in  the  middle. 

For  beams  of  a  similar  description,  loaded  uniformly,  the  rule  is 
the  same,  only  multiply  by  -025  in  place  of  -02. 

RULE. — To  find  the  deflection  of  a  beam  when  fixed  at  one  end 


STRENGTH   OF   MATERIALS. 


295 


and  loaded  at  the  other. — Divide  the  length  in  feet  of  the  fixed  part 
of  the  beam  by  the  length  in  feet  of  the  part  which  yields  to  the 
force,  and  add  1  to  the  quotient ;  then  multiply  the  square  of  the 
length  in  feet  by  the  quotient  so  increased,  and  also  by  *13  ;  divide 
this  product  by  the  middle  depth  in  inches,  and  the  quotient  will 
be  the  deflection,  in  inches  also. 

Multiply  the  deflection  so  obtained  for  cast  iron  by  -86,  the  pro- 
duct equal  the  deflection  for  wrought  iron ;  for  oak,  multiply  by 
.2-8;  and  for  fir,  2-4. 

A  TABLE  of  the  Depths  of  Square  Beams  or  Bars  of  Oast  Iron, 
calculated  to  support  from  1  Cwt.  to  14  Tons  in  the  Middle,  the 
Deflection  not  to  exceed  -^th  of  an  Inch  for  each  Foot  in  Length. 


Lengths  5 

nFeet 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

Weight  ia 
cwt. 

Weigtiu 

i 

| 

i 

,.- 

i 

I 

1 

! 

1 

U 
1 

t 

a 
1 

! 

1 

Tn 

Tn, 

Tn. 

Tn. 

Tn. 

Tn. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

Icwt. 

112 

1-2 

1-4 

1-7 

1-9 

2-0 

2-2 

2-4 

2-5 

2-6 

2-7 

2-9 

3-0 

3-1 

3-2 

2 

124 

V4 

1-7 

2-0 

2-2 

2-4 

2-6 

2-8 

3-0 

3-1 

3-3 

3-4 

3-6 

3-7 

3-8 

3 

336 

V6 

V9 

2-2 

2-4 

2-7 

2-9 

3-1 

3-3 

3-4 

3-6 

3-8 

3-9 

4-1 

4-2 

4 

448 

1-7 

2-0 

2-4 

2-6 

2-9 

3-1 

3-3 

3-5 

3-7 

3-9 

4-0 

4-2 

4-3 

4-5 

5 

560 

1-8 

«•?! 

2-5 

2-8 

3-0 

3-3 

3-5 

3-7 

3-9 

4-1 

4-3 

4-4 

4-6 

4-8 

6 

672 

1-8 

2-2 

2-6 

2-9 

3-2 

3-4 

3-7 

3-9 

4-1 

4-3 

4-5 

4-6 

4-8 

5-0 

7 

784 

V9 

2-3 

2-7 

3-0 

3-3 

3-6 

3-8 

4-1 

4-2 

4-4 

4-6 

4-8 

5-0 

5-2 

8' 

896 

2-0 

?,-4 

2-8 

3-1 

3-4 

3-7 

3-9 

4-2 

4-4 

4-6 

4-8 

5-0 

5-2 

5-4 

9 

1,008 

2-0 

2-5 

2-9 

3-2 

3-5 

3-8 

4-0 

4-3 

4-fl 

4-7 

4-9 

5-1 

5-3 

5-5 

10 

1,120 

2-1 

2-6 

3-0 

3-3 

3-6 

3-9 

4-2 

4-4 

4-7 

4-9 

5-2 

5-3 

6-4 

5-7 

11 

1,232 

2-1 

2-6 

3-0 

3-4 

3-7 

4-0 

4-3 

4-5 

4-8 

5-0 

5-3 

5-4 

6-6 

5-8 

12 

1,344 

2-2 

2-7 

3-1 

3-5 

3-8 

4-1 

4-4 

4-7 

4-9 

6-1 

5-3 

6-5 

5-7 

5-9 

13 

1,456 

2-2 

2-7 

3-1 

3-5 

3-8 

4-2 

4-4 

4-7 

4-9 

5-2 

5-4 

5-6 

5-9 

6-0 

14 

1,568 

2-3 

2-8 

3-2 

3-6 

3-9 

4-2 

4-5 

4-8 

6-0 

5-3 

5-5 

5-7 

6-0 

6-1 

15 

1,680 

2-3 

2-8 

3-2 

3-6 

4-0 

4-3 

4-6 

4-9 

5-2 

5-4 

5-6 

6-8 

6-1 

6-2 

16 

1,792 

2-4 

2-9 

3-3 

3-7 

4-0 

4-4 

4-7 

5-0 

5-2 

5-5 

5-7 

5-9 

6-2 

6-4 

17 

1,904 

2-4 

2-9 

3-4 

3-8 

4-1 

4-4 

4-7 

5-0 

fi-3 

5-5 

5-8 

6-0 

6-2 

6-5 

18 

2,016 

2-4 

3-0 

3-4 

3-8 

4-2 

4-5 

4-8 

n-1 

5-4 

5-6 

5-9 

6-1 

6-4 

6-6 

19 

2,128 

2-5 

3-0 

3-5 

3-9 

4-2 

4-6 

4-9 

fi-2 

5-4 

5-7 

6-0 

6-2 

6-5 

6-7 

1  ton. 

2,240 

2-5 

3-0 

3-5 

3'9 

4-3 

4-6 

4-9 

5-2 

5-5 

6-8 

6-0 

fi-3 

6-ft 

6-8 

If 

2,800 

2-6 

3-2 

3-7 

4-1 

4-5 

4-9 

5-2 

5-5 

5-8 

6-1 

6-4 

6-6 

6-9 

7-2 

i! 

3,360 

2-8 

3-4 

3-9 

4-3 

4-7 

5-1 

5-5 

ft-8 

6-1 

6-4 

6-7 

7-0 

7-2 

7-fi 

i* 

3,920 

2-9 

3-5 

4-0 

4-5 

4-9 

5'3 

5-7 

c-o 

6-3 

6-7 

6-9 

7-2 

7-5 

7-7 

2 

4,480 

2-9 

3-5 

4-1 

4-7 

5-1 

5-5 

5-9 

6-2 

6-5 

6-8 

7-2 

7-6 

T-7 

8-0 

2ft 

5,600 

3-1 

3-8 

4-4 

4-9 

5-5 

5-8 

6-2 

fi-6 

6-9 

7-3 

7-6 

7-9 

8-2 

8-5 

3 

6,720 

3-3 

4-0 

4-6 

5-1 

5-7 

61 

6-5 

fi-q 

7-3 

7-6 

7-9 

8-3 

8-6 

8-9 

3* 

7,840 

3-4 

4-1 

4-8 

5-3 

5-8 

6-3 

6-7 

7-1 

7-6 

7-9 

8-2 

8-6 

8-9 

9-2 

4 

8,960 

3-5 

4-3 

4-9 

5-5 

fi-0 

6-5 

7-0 

7-4 

7-8 

8-2 

8-5 

8-9 

9-2 

9-5 

4i 

10,080 



4-4 

5-1 

5-7 

6-2 

6-7 

7'2 

7-6 

8-0 

8-4 

8-8 

9-1 

9-5 

9-8 

5 

11,200 



4-5 

5-2 

5-8 

6-4 

6-9 

7'4 

7-8 

8-2 

8-6 

9-0 

9-4 

9-7 

10-1 

6 

13,440 

— 

— 

5-5 

6-1 

6-7 

7-2 

7-7 

8-2 

8-6 

9-0 

9-4 

9-8 

10-2 

10-5 

7 

15,680 

— 

— 

5-7 

6-3 

6-9 

7-5 

8-0 

8-ft 

•8-9 

9-4 

9-8 

10-2 

10-6 

11-0 

8 

17,920 





5-9 

6-6 

7-2 

7-8 

8-3 

8-8 

9-3 

9-7 

10-1 

10-6 

10-9 

11-3 

9 

20,160 

— 

— 

6-0 

6-8 

7-4 

8-0 

8-5 

9-0 

9-5 

10-0 

10-4 

10-9 

11-3 

11-7 

10 

22,400 

— 

— 

__ 

6-9 

7-6 

8-2 

8-8 

9-3 

9-8 

ion 

10-7 

11-2 

11-6 

12-0 

11 

24,640 





_ 

7-1 

7-8 

8-4 

9-0 

9-5 

10-0 

10-5 

n-o 

11  -ft 

11-9 

12-3 

12 

26,880 







7-2 

7-9 

8-6 

{»•?, 

9-7 

10-2 

10-8 

ii-fl 

11-7 

12-1 

12-ft. 

13 

29,120 

— 

— 



7-4 

8-1 

8-8 

9-4 

9-9 

10-4 

n-o 

11-5 

11-9 

12-4 

12-8 

14 

31,360 

— 

— 

— 

7-5 

8-3 

8-9 

9-5 

10-1 

10-6 

11-1 

11-7 

12-1 

12-6 

13-0 

Deflection 

in  inches 

•1 

•15 

•2 

•25 

•3 

•35 

•4 

•45 

•5 

•55 

•6 

•65 

•7 

•75 

Length 

in  Feet 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

15 

j    33,600 

7-7 

8-4 

9-1 

9-7 

10-3 

10-8 

11-4 

11-9 

12-3 

12-8 

13-2 

13-7 

14-1 

14-5 

16 

35,840 

7-S 

8-5 

9-2 

9-8 

10-4 

11-0 

11-fi 

12-0 

12-5 

13-0 

13-5 

13-9 

14-3 

14-7 

17 

38,080 

7-9 

8-7 

9-4 

10-0 

10-6 

11-2 

11-7 

12-2 

12-7 

13-2 

13-7 

14-1 

14-5 

14-9 

18 

40,320 

8-0 

8-8 

9-ft 

10-1 

10-8 

11-3 

11-9 

12-4 

12-9 

13-4 

13-9 

14-3 

14-7 

lfi-1 

19 

42,560 

8-1 

8-9 

9-6 

10-3 

10-9 

11-5 

12-2 

12-6 

13-1 

13-6 

14-1 

14-5 

15-0 

15-4 

20 

44,800 

9-0 

9-7 

10-4 

11-0 

11-6 

12-5 

12-7 

13-2 

13-8 

14-2 

14-7 

15-1 

15-fi 

22 

49,280 



9-2 

10-0 

10-7 

11-3 

11-9 

12-8 

13-0 

13-6 

14-1 

14-6 

15-1 

lfi-6 

15-9 

24 

53,760 



9-4 

10-2 

10-9 

11-5 

12-2 

13-0 

13-4 

13-9 

14-4 

14-9 

15-4 

lft-9 

lfi-3 

26 

58,240 



9-fi 

10-4 

11-1 

11-8 

12-4 

13-3 

13-6 

14-2 

14-7 

15-7 

15-7 

16-2 

16-7 

28 

62,720 

— 

9-8 

10-6 

11-4 

12-0 

12-7 

13-5 

13-9 

14-4 

15-0 

15-5 

16-0 

16-5 

17-0 

Deflection 

in  inche* 

•25 

•3 

•35 

•4 

•45 

•5 

•55 

IB 

•66 

'7 

•75 

•8 

•85 

•9 

296 


THE    PRACTICAL   MODEL    CALCULATOR. 


Lengths  in  Feet 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

Weight  in 

Weight  in 

t' 

| 

f' 

t' 

1 

3 

f' 

f- 

t' 

a 

£ 

•g 

* 

a 

tens. 

Ibs. 

1 

& 

I 

i 

i 

i 

I 

In. 

In. 

In. 

In. 

Tn. 

In. 

Tn. 

Tn. 

In. 

Tn. 

Tn. 

Tn. 

Tn 

Tn 

30 

67,200 

10-8 

11-5 

12-2 

12-9 

13-5 

14-1 

14-7 

15-2 

16-7 

lfi-3 

1IV8 

17-3 

17-7 

32 

71,680 

n-o 

11-7 

12-4 

13-1 

13-7 

14-3 

14-9 

15-5 

16-0 

16-ft 

17-0 

17-5 

18-0 

18-5 

34 

76,160 

11-1 

11-9 

12-6 

13-3 

13-9 

14-5 

15-1 

15-7 

16-2 

16-8 

17-3 

17-8 

18-3 

18-8 

36 

80,640 

11-3 

12-0 

12-8 

13-4 

14-1 

14-7 

15-3 

15-9 

16-5 

17-0 

17-5 

18-0 

18-5 

19-0 

38 

85,120 

11-4 

12-2 

13-0 

13-fi 

14-3 

14-9 

1.V5 

lfi-1 

16-7 

17-2 

17-8 

18  -S 

18-8 

19-3 

40 

89,600 



12-4 

13-1 

13-8 

14-5 

15-1 

15-7 

16-4 

16-9 

17-5 

18-0 

18-5 

19-5 

42 

94,080 



12-5 

13-3 

14-0 

14-7 

15-3 

15-9 

16-5 

17-1 

17-7 

18-2 

18-7 

19-3 

19-8 

44 

98,560 



12-7 

13-5 

14-'2 

14-9 

15-5 

lfi-1 

I(v8 

17-4 

17-9 

18-5 

19-0 

19-5 

20-0 

46 

103,040 

— 

12-8 

13-6 

14-3 

15-0 

15-7 

16-3 

17-0 

17T, 

18-1 

18-7 

19-2 

19-8 

20-3 

48 

107,520 

_ 

13-0 

13-7 

14-5 

15-2 

15-9 

16-5 

17-1 

17-7 

18-3 

18-S 

19-4 

20-0    20-5 

50 

112,000 



__ 

13-8 

14-fi 

1.V3 

1fi-0 

IfHi 

17-3 

17-9 

18-5 

19-0 

19-6 

20-1 

'?(V7 

52 

116,480 

— 

— 

14-0 

14-7 

155 

16-2 

16-8 

17-5 

18-1 

18-7 

19-2 

19-8 

20-3 

21-0 

54 

120,960 



— 

14-1 

14-9 

15-7 

16-3 

17-0 

17-6 

18-2 

18-8 

19-4 

19-9 

20-6 

21  -1 

56 

125,440 





14-3 

15-0 

15-8 

16-5 

17-1 

17-8 

18-4 

19-0 

19-6 

20-1 

20-7 

«1-3 

58 

129,920 





144 

15-1 

15-9 

17-3 

17-9 

18-5 

19-2 

19-7 

20-3 

20-9 

21-4 

60 

134,400 

— 

— 

14-5 

15-3 

16-0 

16-7 

17-4 

18-1 

18-7 

19-3 

19-9 

20-5 

21-1 

21-6 

Deflection  in  inches 

•35 

•4  I    -45 

•5 

•55 

•6 

•65 

•7 

•75 

•8 

•85 

•9 

•95 

1-0 

Examples  illustrative  of  the  Table.  —  1.  To  find  the  depth  of  a 
rectangular  bar  of  cast  iron  to  support  a  weight  of  10  tons  in  the 
middle  of  its  length,  the  deflection  not  to  exceed  Jj  of  an  inch  per 
foot  in  length,  and  its  length  20  feet,  also  let  the  depth  be  6  times 
the  breadth. 

Opposite  6  times  the  weight  and  under  20  feet  in  length  is  15-3 
inches,  the  depth,  and  J  of  15-3  =  2-6  inches,  the  breadth. 

2.  To  find  the  diameter  for  a  cast  iron  shaft  or  solid   cylinder 
that  will  bear  a  given  pressure,  the  flexure  in  the  middle  not  to  ex- 
ceed ^th  of  an  inch  for  each  foot  of  its  length,  the  distance  of  the 
bearings  being  20  feet,  and  the  pressure  on  the  middle  equals  10 
tons. 

Constant  multiplier  1-7  for  round  shafts,  then  10  X  1-7  =  17. 
And  opposite  17  tons  and  under  20  feet  is  11-2  inches  for  the  di- 
ameter. 

But  half  that  flexure  is  quite  enough  for  revolving  shafts  :  hence 
17  x  2  =  34  tons,  and  opposite  34  tons  is  13-3  inches  for  the  di- 
ameter. 

3.  A  body  256  Ibs.  weight,  presses  against  its  horizontal  sup- 
port, so  that  it  requires  the  force  of  52  Ibs.  to  overcome  its  friction  ; 
if  the  body  be  increased  to  8750  Ibs.,  what  force  will  cause  it  to 
pass  from  a  state  of  rest  to  one  of  motion  ? 

52 

=  "203125  =  ,  in  this  case,  the  coefficient  of  friction; 


.-.  8750  x  203125  =  1777-34375  Ibs.,  the  force  required. 

This  calculation  is  based  upon  the  law,  that  friction  is  propor- 
tional to  the  normal  pressure  between  the  rubbing  surfaces.  Twice 
the  pressure  gives  twice  the  friction  ;  three  times  the  pressure  gives 
three  times  the  friction  ;  and  so  on.  With  light  pressures,  this  law 
may  not  hold,  but  then  it  is  to  be  attributed  to  the  proportionately 
greater  effect  of  adhesion. 

4.  If  a  sleigh,  weighing  250  Ibs.,  requires  a  force  of  28  Ibs.  to 
draw  it  along  ;  when  1120  Ibs.  are  placed  in  it,  required  the  units 
of  work  expended  to  move  the  whole  350  feet  ? 


STRENGTH    OF   MATERIALS. 


297 


28 
(Tr7r  =  -112,  the  coefficient  of  friction. 

^yOU 

Then  (1120  +  250)  x  -112  =  15344  Ibs.,  the  force  required  to 
move  the  whole. 

.-.  15344  X  350  =  53704,  the  units  of  work  required. 

A  UNIT  OF  WORK  is  the  labour  which  is  equal  to  that  of  raising 
one  pound  a  foot  high.  It  is  supposed  that  a  horse  can  perform 
33000  units  of  work  in  a  minute. 

It  may  also  be  remarked  that  friction  is  independent  of  the  ex- 
tent of  the  surfaces  in  contact,  except  with  trifling  pressures  and 
large  surfaces,  which  is  on  account  of  the  effect  of  adhesion.  The 
friction  of  motion  is  independent  of  velocity,  and  is  generally  less 
than  that  of  quiescence.  A 

5.  Required  the  co- 
efficient of  friction,  for 
a  sliding  motion,  of 
cast  iron  upon  wrought, 
lubricated  with  Dev- 
lin's oil,  and  under 
the  following  circum- 
stances :  the  load  A, 
and  sledge  nin,  weighs 
8420  Ibs.,  and  requires 
a  weight  W,  of  1200  Ibs.  to  cause  it  to  pass  from  a  state  of  rest 
into  one  of  motion  :  the  sledge  and  load  pass  over  22  feet  on  the 
horizontal  way  rs,  in  8  seconds. 

In  this  case  the  coefficient  of  sliding  motion  will  be 
1200  _  1200  +  8420      2  x  22 
8420  "  8420         x  g  x  82' 

in  which  g  =  32-2  feet  ;  the  acceleration  of  the  free  descent  of 
bodies  brought  about  by  gravity.     The  above  expression  becomes 

44 
•142515  -  1-142515  x  =  *11812L 


Hence  the  coefficient  of  the  friction  of  motion  is  '118121,  and  the 
coefficient  of  the  friction  of  quiescence  is  -142515. 

OP   FRICTION,   OR  RESISTANCE  TO   MOTION   IN   BODIES  ROLLING   OR   RUB- 
BING  ON   EACH   OTHER. 

In  the  years  1831,  1832,  and  1833,  a  very  extensive  set  of  ex- 
periments were  made  at  Metz,  by  M.  Morin,  under  the  sanction 
of  the  French  government,  to  determine  as  nearly  as  possible  the 
laws  of  friction  ;  and  by  which  the  following  were  fully  established  : 

1.  When  no  unguent  is  interposed,  the  friction  of  any  two  sur- 
faces (whether  of  quiescence  or  of  motion)  is  directly  proportional 
to  the  force  with  which  they  are  pressed  perpendicularly  together  ; 
so  that  for  any  two  given  surfaces  of  contact  there  is  a  constant 
ratio  of  the  friction  to  the  perpendicular  pressure  of  the  one  surface 
upon  the  other.  Whilst  this  ratio  is  thus  the  same  for  the  same 


298  THE    PRACTICAL    MODEL   CALCULATOR. 

surfaces  of  contact,  it  is  different  for  different  surfaces  of  contact. 
The  particular  value  of  it  in  respect  to  any  two  given  surfaces  of 
contact  is  called  the  coefficient  of  friction  in  respect  to  those  sur- 
faces. 

2.  When  no  unguent  is  interposed,  the  amount  of  the  friction  is, 
in  every  case,  wholly  independent  of  the  extent  of  the  surfaces  of 
contact ;  so  that,  the  force  with  which  two  surfaces  are  pressed  to- 
gether being  the  same,  their  friction  is  the  same,  whatever  may  be 
the  extent  of  their  surfaces  of  contact. 

3.  That  the  friction  of  motion  is  wholly  independent  of  the  velo- 
city of  the  motion. 

4.  That  where  unguents  are  interposed,  the  coefficient  of  friction 
depends  upon  the  nature  of  the  unguent,  and  upon  the  greater  or 
less  abundance  of  the  supply.     In  respect  to  the  supply  of  the  un- 
guent, there  are  two  extreme  cases,  that  in  which  the  surfaces  of 
contact  are  but  slightly  rubbed  with  the  unctuous  matter,  as,  for 
instance,  with  an  oiled  or  greasy  cloth,  and  that  in  which  a  con- 
tinuous stratum  of  unguent  remains  continually  interposed  between 
the  moving  surfaces ;  and  in  this  state  the  amount  of  friction  is 
found  to  be  dependent  rather  upon  the  nature  of  the  unguent  than 
upon  that  of  the  surfaces  of  contact.     M.  Morin  found  that  with 
unguents  (hog's  lard  and  olive  oil)  inteVposed  in  a  continuous  stra- 
tum between  surfaces  of  wood  on  metal,  wood  on  wood,  metal  on 
wood,  and  metal  on  metal,  when  in  motion,  have  all  of  them  very 
near  the  same  coefficient  of  friction,  being  in  all  cases  included  be- 
tween -07  and  -08. 

The  coefficient  for  the  unguent  tallow  is  the  same,  except  in  that 
of  metals  upon  metals.  This  unguent  appears  to  be  less  suited  for 
metallic  substances  than  the  others,  and  gives  for  the  mean  value 
of  its  coefficient,  under  the  same  circumstances,  '10.  Hence,  it  is 
evident,  that  where  the  extent  of  the  surface  sustaining  a  given 
pressure  is  so  great  as  to  make  the  pressure  less  than  that  which 
corresponds  to  a  state  of  perfect  separation,  this  greater  extent  of 
surface  tends  to  increase  the  friction  by  reason  of  that  adhesiveness 
of  the  unguent,  dependent  upon  its  greater  or  less  viscosity,  whose 
effect  is  proportional  to  the  extent  of  the  surfaces  between  which 
it  is  interposed. 

It  was  found,  from  a  mean  of  experiments  with  different  unguents 
on  axles,  in  motion  and  under  different  pressures,  that,  with  the 
unguent  tallow,  under  a  pressure  of  from  1  to  5  cwt.,  the  friction 
did  not  exceed  ^th  of  the  whole  pressure ;  when  soft  soap  was  ap- 
plied, it  became  ^th ;  and  with  the  softer  unguents  applied,  such 
as  oil,  hog's  lard,  &c.,  the  ratio  of  the  friction  to  the  pressure  in- 
creased ;  but  with  the  harder  unguents,  as  soft  soap,  tallow,  and 
anti-attrition  composition,  the  .  friction  considerably  diminished ; 
consequently,  to  render  an  unguent  of  proper  efficiency,  the  nature 
of  the  unguent  must  be  measured  by  the  pressure  or  weight  tend- 
ing to  force  the  surfaces  together. 


STRENGTH    OF    MATERIALS. 


299 


TABLE  of  the  Results  of  Experiments  on  the  Friction  of  Unctuous 
Surfaces.     By  M.  MORIN. 


Surfaces  of  Contact. 


Oak  upon  oak,  the  fibres  being  parallel  to  the  motion  0-018         0-390 
Ditto,  the  fibres  of  the  moving  body  being  perpendicu- 
lar to  the  motion 0-143          0-314 

Oak  upon  elm,  fibres  parallel 0-136 

Elm  upon  oak,           do 0-119         0-420 

Beech  upon  oak,        do 0-330 

Elm  upon  elm,           do 0-140 

Wrought  iron  upon  elm,  do 0-138 

Ditto  upon  wrought  iron,  do 0-177 

Ditto  upon  cast  iron,  do ...             0-118 

Cast  iron  upon  wrought  iron,  do 0-143 

Wrought  iron  upon  brass,  do 0-160 

Brass  upon  wrought  iron,  do 0-166 

Cast  iron  upon  oak,  do 0-107          0-100 

Ditto  upon  elm,  do.,  the  unguent  being  tallow 0-125 

Ditto,  do.,  the  unguent  being  hog's  lard  and  black 

lead 0-137 

Elm  upon  cast  iron 0-135          0-098 

Cast  iron  upon  cast  iron 0-144 

Ditto  upon  brass 0-f32 

Brass  upon  cast  iron 0-107 

Ditto  upon  brass 0-134         0-164 

Copper  upon  oak 0-100 

Yellow  copper  upon  cast  iron 0-115 

Leather  (ox-hide),  well  tanned,  upon  cast  iron,  wetted  0-229          0-267 

Ditto  upon  brass,  wetted 0-244 


Coefficients  of  Friction. 


Friction  of      Friction  of 
Motion.        Quiescence. 


In  these  experiments,  the  surfaces,  after  having  been  smeared 
with  an  unguent,  were  wiped,  so  that  no  interposing  layer  of  the 
unguent  prevented  intimate  contact. 

TABLE  of  the  Results  of  Experiments  on  Friction,  with  Unguents 
interposed.     By  M.  MORIN. 


Surfaces  of  Contact. 


Coefficients  of  Friction. 


Friction  of       Friction  of 
Motion.         Quiescence. 


Unguents. 


Oak  upon  oak,  fibres  parallel.... 

Do.  do 

Do.  do 

Do.,  fibres  perpendicular 

Do.  do 

Do.  do 

Do.  upon  elm,  fibres  parallel 

Do.  do 

Do.  do 

Do.  upon  cast  iron 

Do.  upon  wrought  iron 

Beech  upon  oak,  fibres  parallel.. 

Elm  upon  oak,    do 

Do.  do 

Do.  do 

Elm  upon  elm,    do 

Do.  upon  cast  iron 

Wrought  iron  upon  oak,  fibres  \ 

parallel / 

Do.  do 


0-164 
0-075 
0-067 
0-083 
0-072 
0-250 
0-136 
0-073 
0-066 
0-080 
0-098 
0-055 
0-137 
0-170 
0-060 
0-139 
0-066 

0-256 
0-214 


0-440 

0-164 

0-254 


0-178 


0-411 
0-142 

0-217 


0-649 


Dry  soap. 
Tallow. 
Hog's  lard. 
Tallow. 
Hog's  lard. 
Water. 
Dry  soap. 
Tallow. 
Hog's  lard. 
Tallow. 
Tallow. 
Tallow. 
Dry  soap. 
Tallow. 
Hog's  lard. 
Dry  soap. 
Tallow. 

f  Greased  and  satu- 
\  rated  with  water. 
Dry  soap. 


300 


THE  PRACTICAL  MODEL  CALCULATOR. 


Surfaces  of  Contact. 

Coefficients  of  Friction. 

Unguents. 

Tallow. 

Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Dry  soap. 
f  Greased  and  satu- 
\  rated  with  water. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Olive  oil. 
f  Hog's  lard  and 
\  plumbago. 
Tallow. 
Water. 
Soap. 
Tallow. 
Hog's  lard. 
Olive  oil. 
f  Hog's  lard  and 
\  plumbago. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Olive  oil. 
Tallow. 
f  Lard  and  plum- 
\  bago. 
Olive  oil. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Tallow. 
Olive  oil. 
f  Laf  d  and  plum- 
\  bago. 
f  Greased  and  satu- 
\  rated  with  water. 

Friction  of 
Motion. 

Friction  of 
Quiescence. 

Wrought  iron  upon  oak,  fibres  \ 

0-085 

0-078 
0-076 
0-055 
0-103 
0-076 
0-066 
0-082 
0-081 
0-070 
0-103 
0-075 
0-078 
0-189 

0-218 

0-078 
0-075 
0-075 
0-077 
0-061 

0-091 

0-108 

o-i  bo 

0-115 

0-646 
0-100 

o-'ibo 

0-100 

o-i  bo 

0-100 

o-'ibo 

0-103 
0-106 

0-108 

Do  upon  elm      do  

Do                       do  

Do                        do 

Do  upon  cast  iron  do  

Do                        do    

Do                        do 

Do  upon  wrought  iron   do       . 

Do                       do:  

Do                       do  

Wrought  iron  upon  brass,  do  
Do                        do             

Do                       do  

Do                       do 

Do                       do  

Do                       do  

Do                       do               ...     . 

Do  upon  elm      do    

0-314 
0-197 
0-100 
0-070 
0-064 

0-055 

0-103 
0-075 
0-078 
0-069 
0-072 
0-068 
0-066 
0-086 
0-077 
0-081 

0-089 

0-072 
0-058 
0-105 
0-081 
0-079 
0-093 
0-076 
0-056 
0-053 

0-067 
0-365 

Do                    do            .         

Do                    do      ..  ..*  

Do                    do               

Do             do               

Do             do  >  

Copper  upon  oak,  fibres  paralle' 
Yellow  copper  upon  cast  iron.... 
Do                                     do  

Do.                                    do.. 
Brass  upon  cast  iron    • 

Do                     do     

Do                     do     

Do                     do  

Steel  upon  cast  iron        

Do                    do     

Do  upon  wrought  iron            «... 

Do                    do    Tl  

Do.            do  

Do            do       

Tanned  ox-hide  upon  cast  iron.. 

The  extent  of  the  surfaces  in  these  experiments  bore  such  a  relation  to  the  pres- 
sure as  to  cause  them  to  be  separated  from  one  another  throughout  by  an  inter- 
posed stratum  of  the  unguent. 


STRENGTH   OF   MATERIALS. 


301 


TABLE  of  the  Results  of  Experiments  on  the  Friction  of  dudgeons 
or  Axle-ends,  in  motion  upon  their  bearings.     By  M.  MOBIX. 


Surfaces  in  Contact. 

State  of  the  Surfaces. 

Coefficient  of  Friction. 

Cast    iron    axles    in 
cast  iron  bearings. 

Cast    iron    axles    in 
cast  iron  bearings. 

Wrought  iron   axles 
in  cast  iron  bear- 
ings. 

Wrought   iron   axles 
in  brass  bearings. 

Iron  axles  in  lignum 
vitge  bearings. 

Brass  axles  in  brass 
bearings. 

"  Coated  with  oil  of  olives,  ") 
with  hog's  lard,  tallow,  > 
and  soft  gome  J 

0-07  to  0-08 

0-08 
0-054 
0-14 
0-14 

0-07  to  0-08 

0-16 
0-16 
0-19 

0-07  to  0-08 

0-07  to  0-08 

0-09 
0-19 
0-25 
0-11 

0-19 
0-10 
0-09 

-  With  the  same  and  water... 
Coated  with  asphaltum  

Greasy  and  wetted 

"  Coated  with  oil  of  olives,  \ 
with  hog's  lard,  tallow,  I 
and  soft  gome  J 

"1  Greasy 

Scarcely  greasy  

f  Coated  with  oil  of  olives,  ~) 
•j      tallow,  hog's  lard,  or  L 

r  Coated  with  oil  of  olives,  \ 
hog's  lard,  or  tallow,    / 
-]  Coated  with  hard  gome  
Greasy  and  wetted  

[Scarcely  greasy  

C  Coated  with  oil  or  hog's  ) 
1      lard     .                            } 

1  Greasy     

f  Coated  with  oil'  

\  With  hog's  lard  

TABLE  of  Coefficients  of  Friction  under  Pressures  increased  continu- 
ally up  to  limits  of  Abrasion. 


Pressure  per 
Square  Inch. 

Coefficients  of  Friction. 

Wrought  Iron  upon 
Wrought  Iron. 

Wrought  Iron  upon 
Cast  Iron. 

Steel  upon  Cast 
Iron. 

Brass  upon  Cast 
Iron. 

32-51bs. 

•140 

•174 

•166 

•157 

l-66cwts. 

•250 

•275 

•300 

•225 

2-00 

•271 

•292 

•333 

•219 

2-33 

•285 

•321 

•340 

•214 

2-66 

•297 

•329 

-      -344 

•211 

3-00 

•312 

•333 

•347 

•215 

3-33 

•350 

•351 

•351 

•206 

3-66 

•376 

•353 

•353 

•205 

4-00 

•395 

•365 

•354 

•208 

4-33 

•403 

•366 

•356 

•221 

4-66 

•409 

•366 

•357 

•223 

5-00 



•367 

•358 

•233 

5-33 

. 

•367 

•359 

•234 

5-66 

•367 

•367 

•235 

6-00 

•376 

•403 

•233 

6-33 

. 

•434 



•234 

,       6-66 

. 





•235 

7-00 

. 





•232 

7-33 

.    .. 



•273 

2  A 


302  THE   PRACTICAL   MODEL    CALCULATOR. 

Comparative  friction  of  steam  engines  of  different  modifications, 
if  the  beam  engine  be  taken  as  the  standard  of  comparison  : — 

The  vibrating  engine has  a  gain  of  1*1  per  cent. 

The  direct-action  engine,  with  slides  —     loss  of  1-8     — 

Ditto,  with  rollers —     gain  of  0'8     — 

Ditto,  with  a  parallel  motion —     gain  of  1'3     — 

Excessive  allowance  for  friction  has  hitherto  been  made  in  cal- 
culating the  effective  power  of  engines  in  general ;  as  it  is  found 
practically,  by  experiments,  that,  where  the  pressure  upon  the  pis- 
ton is  about  12  Ibs.  per  square  inch,  the  friction  does  not  amount 
to  more  than  1J  Ibs. ;  and  also  that,  by  experiments  with  an  indi- 
cator on  an  engine  of  50  horse  power,  the  whole  amount  of  friction 
did  not  exceed  5  horse  power,  or  one-tenth  of  the  whole  power  of 
the  engine. 

RECENT  EXPERIMENTS  MADE  BY  M.  MORIN  ON  THE  STIFFNESS  OF  ROPES, 
OR  THE  RESISTANCE  OF  ROPES  TO  BENDING  UPON  A  CIRCULAR  ARC. 

The  experiments  upon  which  the  rules  and  table  following  are 
founded  were  made  by  Coulomb,  with  an  apparatus  the  invention 
of  Amonton,  and  Coulomb  himself  deduced  from  them  the  follow- 
ing results : — 

1.  That  the  resistance  to  bending  could  be  represented  by  an 
expression  consisting  of  two  terms,  the  one  constant  for  each  rope 
and  each  roller,  which  we  shall  designate  by  the  letter  A,  and 
which  this  philosopher  named  the  natural  stiffness,  because  it  de- 
pends on  the  mode  of  fabrication  of  the  rope,  and  the  degree  of 
tension  of  its  yarns  and  strands ;  the  other,  proportional  to  the 
tension,  T,  of  the  end  of  the  rope  which  is  being  bent,  and  which 
is  expressed  by  the  product,  BT,  in  which  B  is  also  a  number 
constant  for  each  rope  and  each  roller. 

2.  That  the  resistance  to  bending  varied  inversely  as  the  diame- 
ter of  the  roller. 

Thus  the  complete  resistance  is  represented  by  the  expression 

A  +  BT 
~D ' 

where  D  represents  the  diameter  of  the  roller. 

Coulomb  supposed  that  for  tarred  ropes  the  stiffness  was  pro- 
portional to  the  number  of  yarns,  and  M.  Navier  inferred,  from 
examination  of  Coulomb's  experiments,  that  the  coefficients  A  and 
B  were  proportional  to  a  certain  power  of  the  diameter,  which  de- 
pended on  the  extent  to  which  the  cords  were  worn.  M.  Morin, 
however,  deems  this  hypothesis  inadmissible,  and  the  following  is 
an  extract  from  his  new  work,  "Lesons  de  Me'canique  Pratique," 
December,  1846  :— 

"  To  extend  the  results  of  the  experiments  of  Coulomb  to  ropes 
of  different  diameters  from  those  which  had  been  experimented 
upon,  M.  Navier  has  allowed,  very  explicitly,  what  Coulomb  had 
but  surmised :  that  the  coefficients,  A,  were  proportional  to  a  cer- 


STRENGTH    OF   MATERIALS. 


303 


tain  power  of  the  diameter,  which  depended  on  the  state  of  wear 
of  the  ropes ;  but  this  supposition  appears  to  us  neither  borne  out, 
nor  even  admissible,  for  it  would  lead  to  this  consequence,  that  a 
worn  rope  of  a  metre  diameter  would  have  the  same  stiifness  as  a 
new  rope,  which  is  evidently  wrong ;  and,  besides,  the  comparison 
alone  of  the  values  of  A  and  B  shows  that  the  power  to  which  the 
diameter  should  be  raised  would  not  be  the  same  for  the  two  terms 
of  the  resistance." 

Since,  then,  the  form  proposed  by  M.  Navier  for  the  expression 
of  the  resistance  of  ropes  to  bending  cannot  be  admitted,  it  is  ne- 
cessary to  search  for  another,  and  it  appears  natural  to  try  if  the 
factors  A  and  B  cannot  be  expressed  for  white  ropes,  simply  accord- 
ing to  the  number  of  yarns  in  the  ropes,  as  Coulomb  has  inferred 
for  tarred  ropes. 

Now,  dividing  the  values  of  A,  obtained  for  each  rope  by  M. 
Navier,  by  the  number  of  yarns,  we  find  for 


n  =  30  d  =  Om-200  A 


0-222460  -  =  0-0074153. 
n 


n  =  15  d  =  Om-144  A  =  0-063514  -  =  0-0042343. 

n  =  6  d  =  Om-0088  A  =  0-010604  -  =  0-0017673. 

n 

It  is  seen  from  this  that  the  number  A  is  not  simply  propor- 
tional to  the  number  of  yarns. 

Comparing,  then,  the  values  of  the  ratio  —  corresponding    to 
the  three  ropes,  we  find  the  following  results : — 


Differences  of 

Number  of 

Values  of 
A 

Differences  of  the  numbers  of 

Differences  of 
the  values  of 

the  values  of 
—  for  each 

yarns. 

n 

3'arns. 

n' 

yarn  of 

difference. 

30 

0-0074153 

From  30  to  15.     15  yarns 

0-0031810 

0-000212 

15 

0-0042343 

—    15  to    6.       9     — 

0-0024770 

0-000272 

6 

0-0017673 

—   30  to    6.     24    — 

0-0056400 

0-000252 

Mean  difference  per  yarn,  0-000245 

It  follows,  from  the  above,  that  the  values  of  A,  given  by  the 
experiments,  will  be  represented  with  sufficient  exactness  for  all 
practical  purposes  by  the  formula 

A  =  n  [0-0017673  +  0-000245  (n  -—  6)]. 
=  n  [0-0002973  +  0-000245  *| 

An  expression  relating  only  to  dry  white  ropes,  such  as  were  used 
by  Coulomb  in  his  experiments. 

With  regard  to  the  number  B,  it  appears  to  be  proportional  to 
the  number  of  yarns,  for  we  find  for 


•304  THE   PRACTICAL   MODEL   CALCULATOR. 

n  =  30  d  =  Om-0200  B  =  0-009738  -  =  0-0003246 

n 

T> 

n  =  15  d  =  Om-0144  B  =  0-005518  -  =  0-0003678 

n 

n  =  6  d  =  Om-0088  B  =  0-002380  -  =  0-0003967 

n 

Mean 0-0003630 

Whence 

B  =  0-000363  n. 

Consequently,  the  results  of  the  experiments  of  Coulomb  on  dry 
white  ropes  will  be  represented  with  sufficient  exactness  for  prac- 
tical purposes  by  the  formula 

K  =  n  [0-000297  -f  0-000245  n  +  0-000363  T]  kil. 
which  will  give  the  resistance  to  bending  upon  a  drum  of  a  metre 
in  diameter,  or  by  the  formula 

R  =  g.  [0-000297  +  0-000245  n  +  0-000363  T]  kil. 

for  a  drum  of  diameter  D  metres. 

These  formulas,  transformed  into  the  American  scale  of  weights 
and  measures,  become 

B,  =  n  [0-0021508  -f-  0-0017724  n  +  0-00119096  T]  Ibs. 
for  a  drum  of  a  foot  in  diameter,  and 

R  =  p  [0-0021508  +  0-0017724  n  4-  0.00119096  T]  Ibs. 

for  a  drum  of  diameter  D  feet. 

With  respect  to  worn  ropes,  the  rule  given  by  M.  Navier  cannot 
be  admitted,  as  we  have  shown  above,  because  it  would  give  for  the 
stiffness  of  a  rope  of  a  diameter  equal  to  unity  the  same  stiffness 
as  for  a  new  rope. 

The  experiments  of  Coulomb  on  worn  ropes  not  being  sufficiently  • 
complete,  and  not  furnishing  any  precise  data,  it  is  not  possible, 
without  new  researches,  to  give  a  rule  for  calculating  the  stiffness 
of  these  ropes. 

TARRED   ROPES. 

In  reducing  the  results  of  the  experiments  of  Coulomb  on  tarred 
ropes,  as  we  have  done  for  white  ropes,  we  find  the  following 
values : — 

n  =  30  yarns'  A  =  0-34982        B  =  0-0125605 
n  =  16     —    A  =  0-106003       B  =  0-006037 
n  =    6     —    A  =  0-0212012     B  =  0-0025997 

which  differ  very  slightly  from  those  which  M.  Navier  has  given. 
But,  if  we  look  for  the  resistance  corresponding  to  each  yarn,  we 
find 


STRENGTH    OF    MATERIALS. 


305 


B 


n 

n 
n 

—  30  yarns 
=  15  — 
=  6 

n 
A 
n 
A 
n 

=  0 
=  0 
=  0 

•0116603 
•0070662 
•0035335 

n 
B 
n 
B 

n 

=  0 
=  0 
=  0 

•000418683 
•000402466 
•000433283 

Mean 0-000418144 

We  see  by  this  that  the  value  of  B  is  for  tarred  ropes,  as  for 
ropes,  sensibly  proportional  to  the  number  of  yarns,  but  it 
is  not  so  for  that  of  A,  as  M.  Navier  has  supposed. 

A 

Comparing,  as  we  have  done  for  white  ropes,  the  values  of  — 

corresponding  to  the  three  ropes  of  30,  15,  and  6  yarns,  we  obtain 
the  following  results  : — 


Number  of 
yarns. 

Values  of 

A 

n 

Differences  of  the  number  of 
yarns. 

Differences  of 
the  values  of 
A 
n 

Differences  of 
the  values  of 

—  for  each 
n 
yarn  of 
difference. 

30  " 
15 

6 

0-0116603 
0-0070662 
0-0035335 

From  30  to  15.     15  yarns 
—      15  to    6.      4    — 
—      60  to    6.     25    — 

0-0045941 
0-0035327 
0-0081268 

0-000306 
0-000392 
0-000339 

Mean 0-000346 

It  follows  from  this  that  the  value  of  A  can  be  represented  by 
the  formula 

A  =  n  [0-0035335  +  0-000346  (n  -  6)1 

=  n  [0-0014575  +  0-000346  ri] 
and  the  whole  resistance  on  a  roller  of  diameter  D  metres,  by 

R  =  5  [0-0014575  +  0-000346  n  +  0-000418144  T]  kil. 

Transforming  this  expression  to  the  American  scale  of  weights  and 
measures,  we  have 

R  =  p  [0-01054412  +  0-00250309  n  +  0-001371889  T]  Ibs. 

for  the  resistance  on  a  roller  of  diameter  D  feet. 

This  expression  is  exactly  of  the  same  form/  as  that  which  relates 
to  white  ropes,  and  shows  that  the  stiffness  of  tarred  ropes  is  a  little 
greater  than  that  of  new  white  ropes. 

In  the  following  table,  the  diameters  corresponding  to  the  differ- 
ent numbers  of  yarns  are  calculated  from  the  data  of  Coulomb,  by 
the  formulas, 

d  cent.  =  \/0-1338  n  for  dry  white  ropes,  and 
d  cent.  =  -v/0.186  n  for  tarred  ropes, 
which,  reduced  to  the  American  scale,  become 

d  inches  =  ^0-020739  n  for  dry  white  ropes,  and 
d  inches  =  v/0 -02883  for  tarred  ropes. 
2A2  20 


306 


THE  PRACTICAL  MODEL  CALCULATOR. 


NOTE.  — The  diameter  of  the  rope  is  to  be  included  in  D  ;  thus, 
with  an  inch  rope  passing  round  a  pulley,  8  inches  in  diameter  in 
the  groove,  the  diameter  of  the  roller  is  to  be  considered  as  9 
inches. 


i 
c 

Dry  White  Ropes. 

Tarred  Ropes. 

•5 

0 

z 

Diameter. 

Value  of  the  natural 
stiffness,  A. 

Value  of  the  stiff- 
ness proportional 
to  the  tension,  B. 

Diameter. 

Value  of  the  natural 
stiffness,  A. 

Value  of  the  stiff- 
ness  proportional 
to  the  tension,  li. 

ft. 

Ibs. 

I!        ft. 

Ibs. 

6 

0-0293 

0-0767120 

0-0071457 

0-0347 

0-153376 

0-00823133 

9 

0.0360 

0-1629234 

0-0107]  86 

0-0425 

0-297647 

0-01  2.^700 

12 

0-0416 

0-2810384 

0-0142915 

0-0490 

0-486976 

0-01646267 

15 

0-0465 

0-4310571 

0-0178644 

0-0548 

0-721357 

0-02057834 

18 

0-0509 

0-6129795 

0-0214373 

0-OWJO 

0-000795 

0-02469400 

21 

0-0550 

0-8268054 

0-0250102 

0-0648 

1-325289 

0-02880967 

24 

0-0588 

1.0725350 

0-0285831 

0-0693 

1-694839 

0-03292534 

27 

0-0622 

1-3501082 

0-0321559 

0-0735 

2-109444 

0-03704100 

30 

0-0657 

1-6597051 

0-0357288 

0-0775 

2-569105 

0-04115667 

33 

0-0689 

2-0011455 

0-0393017 

0-0813 

3-073821 

0-04527234 

36 

0-0720 

2-3744897 

00428746 

0-0849 

3-623593 

0-04938SOO 

0-0749 

2-7797375 

0-0464475 

6-0884 

4-218416 

0-05350367 

42 

0-0778 

3-216S8S8 

0-0500203 

0-0917 

4-858304 

0-05761934 

45 

0-0805 

3-6859433 

0-0535932 

0-0949 

5-543242 

0-06173501 

48 

0-0831 

4-1869024 

0-0571661 

0-0980 

6-273237 

0-06585067 

51 

0-0857 

4-7197647 

0-0607390 

0-1010 

7-048287 

0-06996634 

54 

0-0882 

5-2845306 

0-0643119 

0-1040 

7-868393  ' 

0-07408201 

57 

0-0908 

5-8812001 

0-0678847 

0-1070 

8-733554 

0-07819767 

60 

0-0926 

6-5097733 

0-0714576 

0-1099 

9-643771 

0-08231334 

f     0-0021508W 

(     0-01054412ra 

n 

)/0-000144» 

"j+0-0017724/i|- 

0-001i0096?i 

y/0-00020» 

1+0-00250309?A 

0-001371889n 

Application  of  the  preceding  Tables  or  Formulas. 
To  find  the  stiffness  of  a  rope  of  a  given  diameter  or  number  of 
yarns,  we  must  first  obtain  from  the  table,  or  by  the  formulas,  the 
values  of  the  quantities  A  and  B  corresponding  to  these  given 
quantities,  and  knowing  the  tension,  T,  of  the  end  to  be  wound 
up,  we  shall  have  its  resistance  to  bending  on  a  drum  of  a  foot  in 
diameter,  by  the  formula 

R  =  A  +  BT. 

Then,  dividing  this  quantity  by  the  diameter  of  the  roller  or 
pulley  round  which  the  rope  is  actually  to  be  bent,  we  shall  have 
the  resistance  to  bending  on  this  roller. 

What  is  the  stiffness  of  a  dry  white  rope,  in  good  condition,  of 
60  yarns,  or  -0928  diameter,  which  passes  over  a  pulley  of  6  inches 
diameter  in  the  groove,  under  a  tension  of  1000  Ibs.  ?  The  table 
gives  for  a  dry  white  rope  of  60  yarns,  in  good  condition,  bent 
upon  a  drum  of  a  foot  in  diameter, 

A  =  0-50977     B  =  0-0714576 
and  we  have  D  =  0-5  +  0-0928 ;  and  consequently, 
6-50977  +  0-0714576  x  1000 


R 


128  Ibs. 


0-5928 

The  whole  resistance  to  be  overcome,  not  including  the  friction 
on  the  axis,  is  then 

Q  -f  R  =  1000  +  128  =  1128  Ibs. 

The  stiffness  in  this  case  augments  the  resistance  by  more  than 
one-eighth  of  its  value. 


STRENGTH    OF   MATERIALS.  307 

FURTHER  RECENT  EXPERIMENTS   MADE   BY  M.  MORIN,  ON  THE  TRAC- 

TION  OF   CARRIAGES,  AND   THE    DESTRUCTIVE    EFFECTS   WHICH    THEY 
PRODUCE    UPON    THE    ROADS. 

The  study  of  the  effects  which  are  produced  when  a  carriage  is 
set  in  motion  can  be  divided  into  two  distinct  parts  :  the  traction 
of  carriages,  properly  so  called,  and  their  action  upon  the  roads. 

The  researches  relative  to  the  traction  of  carriages  have  for  their 

object  to  determine  the  magnitude  of  the  effort  that  the  motive 

*  power  ought  to  exercise  according  to  the  weight  of  the  load,  to  the 

diameter  and  breadth  of  the  wheels,  to  the  velocity  of  the  carriage, 

t  and  to  the  state  of  repair  and  nature  of  the  roads. 

The  first  experiments  on  the  resistance  that  cylindrical  bodies 
offer  to  being  rolled  on  a  level  .surface  are  due  to  Coulomb,  who 
determined  the  resistance  offered  by  rollers  of  lignum  vitae  and 
elm,  on  plane  oak  surfaces  placed  horizontally. 

His  experiments  showed  that  the  resistance  was  directly  propor- 
tional to  the  pressure,  and  inversely  proportional  to  the  diameter 
of  the  rollers. 

If,  then,  P  represent  the  pressure,  and  r  the  radius  of  the  roller, 
the  resistance  to  rolling,  R,  could,  according  to  the  laws  of  Cou- 
lomb, be  expressed  by  the  formula 


in  which  A  would  be  a  number,  constant  for  each  kind  of  ground, 
but  varying  with  different  kinds,  and  with  the  state  of  their 
surfaces. 

The  results  of  experiments  made  at  Vincennes  show  that  the 
law  of  Coulomb  is  approximately  correct,  but  that  the  resistance 
increases  as  the  width  of  the  parts  in  contact  diminishes. 

Other  experiments  of  the  same  nature  have  confirmed  these  con- 
clusions ;  and  we  may  allow,  at  least,  as  a  law  sufficiently  exact 
for  practical  purposes,  that  for  woods,  plasters,  leather,  and  gene- 
rally for  hard  bodies,  the  resistance  to  rolling  is  nearly  — 

-1st.  Proportional  to  the  pressure. 

2d.    Inversely  proportional  to  the  diameter  of  the  wheels. 

3d.    Greater  as  the  breadth  of  the  zone  in  contact  is  smaller. 

EXPERIMENTS  UPON  CARRIAGES  TRAVELLING  ON  ORDINARY  ROADS. 

These  experiments  were  not  considered  sufficient  to  authorize 
the  extension  of  the  foregoing  conclusions  to  the  motion  of  car- 
riages on  ordinary  roads.  It  was  necessary  to  operate  directly  on 
the  carriages  themselves,  and  in  the  usual  circumstances  in  which 
they  are  placed.  Experiments  on  this  subject  were  therefore  un- 
dertaken, first  at  Metz,  in  1837  and  1838,  and  afterwards  at  Cour- 
bevoie,  in  1839  and  1841,  with  carriages  of  every  species  ;  and 
attention  was  directed  separately  to  the  influence  upon  the  magni- 
tude of  the  traction,  of  the  pressure,  of  the  diameter  of  the  wheels, 
of  their  breadth,  of  the  speed,  and  of  the  state  of  the  ground. 

In  heavily  laden  carriages,  which  it  is  most  important  to  take 


308 


THE  PRACTICAL  MODEL  CALCULATOR. 


into  consideration,  the  weight  of  the  wheels  may  be  neglected  in 
comparison  with  the  total  load ;  and  the  relation  between  the  load 
and  the  traction,  upon  a  level  road,  is  approximately  given  by  the 
equation — 

F       2  ( A  X  fr  } 

|T=  — r,  x  r,]  for  carriages  with  four  wheels, 


and          ^FT= 


-1  for  carriages  with  two  wheels, 


in  which  F4  represents  the  horizontal  component  of  the  traction ; 
Pj  the  total  pressure  on  the  ground ; 
r'  and  r"  the  radii  of  the  fore  and  hind  wheels ; 
r±  the  mean  radius  of  the  boxes  ; 
/  the  coefficient  of  friction ; 

and  A  the  constant  multiplier  in  Coulomb's  formula  for  the 
resistance  to  rolling. 

These  expressions  will  serve  us  hereafter  to  determine,  by  aid  of 
experiment,  the  ratio  of  the  traction  to  the  load  for  the  most  usual 
cases. 

Influence  of  the  Pressure. 

To  observe  the  influence  of  the  pressure  upon  the  resistance  to 
rolling,  the  same  carriages  were  made  to  pass  with  different  loads 
over  the  same  road  in  the  same  state. 

The  results  of  some  of  these  experiments,  made  at  a  walking  pace, 
are  given  in  the  following  table  : — 


Carriages  employed. 

Road  traversed. 

Pressure. 

Traction. 

Ratio  of  the 
traction  to 
the  load. 

kil. 

kil. 

Chariotporte  corps 

Road  from  Courbe- 

6992 

180-71 

1/38-6 

d'artillerie. 

voie  to  Colomber, 

6140  - 

159-9 

1/39-2 

dry,   in   good  re- 

4580 

113-7 

1/40-2 

pair,  dusty. 

Chariot  deroulage, 

Road  from  Courbe- 

7126 

138-9 

1/51-3 

without  springs. 

voie    to    Bezous, 

5458 

115-5 

1/48-9 

solid,  *hard  gra- 

4450 

93-2 

1/47-7 

vel,  very  dry. 

3430 

68-4 

1/50-2 

Chariot  de  roulage, 

Road  from  Colomber 

1600 

39-3 

1/40-8 

with  springs. 

to        Courbevoie, 

3292 

89-2 

1/36-9 

pitched,  in  ordina- 

4996 

136-0 

1/36-8 

ry  rep  air,  f  muddy 

Carriages  with  six 

Road  from  Courbe- 

3000 

138-9 

1/21-6 

equal  wheels. 

voie  to  Colomber, 

4692 

224-0 

1/21-0 

Two  carriages  with 

deep    ruts,    with 

6000 

285-8 

1/21-0 

six  equal  wheels, 

muddy  detritus. 

6000 

286-7 

1/21-0 

hooked   on,  one 

behind  the  other. 

From  the  examination  of  this  table,  it  appears  that  on  Jsolid 
gravel  and  on  pitched  roads  the  resistance  of  carriages  to  traction 
is  sensibly  proportional  to  the  pressure. 

*  En  gravier  dur.        f  Pave'  en  6tat  ordinaire.        J  En  empierrement  solide. 


STRENGTH    OP   MATERIALS. 


309 


We  remark  that  the  experiments  made  upon  one  and  upon  two 
six-wheeled  carriages  have  given  the  same  traction  for  a  load  of 
6000  kilogrammes,  including  the  vehicle,  whether  it  was  borne 
upon  one  carriage  or  upon  two.  -It  follows  thence  that  the  trac- 
tion is,  cseteris  paribus  and  between  certain  limits,  independent  of 
the  number  of  wheels. 

Influence  of  the  Diameter  of  the  Wheels. 

To  observe  the  influence  of  the  diameter  of  the  wheels  on  the 
traction,  carriages  loaded  with  the  same  weights,  having  wheels 
with  tires  of  the  same  width,  and  of  which  the  diameters  only  were 
varied  between  very  extended  limits,  were  made  to  traverse  the 
same  parts  of  roads  in  the  same  state.  Some  of  the  results  obtained 
are  given  in  the  following  table. 

These  examples  show  that  on  solid  roads  it  may  be  admitted  as 
a  practical  law  that  the  traction  is  inversely  proportional  to  the 
diameters  of  the  wheels.  * 


Diameter  of 

Diameter  of 

. 

a! 

the  wheels  in 

the  wheels  in 

pT 

^H 

Ratio  of 

•S3 

Value 

Value 

. 

Road 

metres. 

English  feet. 

32 

f 

thetrac- 

S3 

Resist- 

of  A 

of  A 

Carnages  emp  oyed. 

raverse  • 

3| 

| 

the  pres- 

Is- 

rolling, 

French 

American 

Fore 

Hind 

Fore 

Hind 

sure. 

R. 

scale. 

scale. 

wheels 

wheels 

wheels 

wheels 

A 

EH 

&l 

2  r' 

2r" 

2r' 

2r" 

1 

m. 

m. 

kil. 

kil. 

kil. 

kil. 

Chariot     porte 

Road  from  Cour- 

2-029  2-029 

6-657 

6-657 

4928 

81-6 

1/60- 

9-6 

72-0 

0-0148 

0-04856 

corps  d'artil- 

bevoie  to  Colom- 

1-453  (1-453 

4-767 

4-767 

4930 

108-6 

1/45-6 

14-4 

94-2 

0'0139 

0-04560 

lerie. 

ber,  *solid  gra- 

0-8720-872 

2-861 

2-861 

4924 

179-0 

1/27-4 

25-3 

153-7 

0-0137 

0-04494 

vel,  dusty. 

Porte  corps  d'ar- 

2-0292-029 

6-657 

6-657 

4692 

51-45 

1/90-45 

9-0 

42-45  0-0092 

0-03018 

tillerie. 

1-4531-453 

4-767 

4-767 

4594 

71-45 

1/64-3 

13-2 

58-250-0092 

0-03018 

Chariot  comtois. 
A     six-wheeled 
carriage. 
The  same  with 

•{•Pitched  pave- 
•  ment  of  Fon- 
tainebleau. 

1-110 
0-860 

1-358 
0-860 

3-642 
2-822 

4-455 
2-822 

1871 
3270 

32-10 

81-05 

1/58-4 
1/40-4 

4-7 
9-7 

27-400-0089 
71-350-0094 

0-02920 
0-03084 

four  wheels. 

0-860  0-860 

2-822 

2-822 

3270 

78-80 

1/41-5 

9-7 

69-100-0091 

0-02986 

Camion. 

0-592  '0-660 

1-942 

2-165 

1500 

52-30 

1/28-8 

8-8 

43-5010-0091 

0-02986 

Camion. 

- 

0-420  ;  0-597 

1-378 

1-959 

1600 

63-20 

1/22-4 

11-6 

56-60  0-0089 

0-02920 

Influence  of  the  Width  of  the  Felloes. 

Experiments  made  upon  wheels  of  different  breadths,  having  the 
same  diameter,  show,  1st,  tkat  on  soft  ground  the  resistance  to 
rolling  increases  as  the  width  of  the  felloe;  2dly,  on  solid  gravel 
and  pitched  roads,  the  resistance  is  very  nearly  independent  of  the 
width  of  the  felloe. 

Influence  of  the  Velocity. 

To  investigate  the  influence  of  the  velocity  on  the  traction  of 
carriages,  the  same  carriages  were  made  to  traverse  different  roads 
in  various  conditions ;  and  in  each  series  of  experiments  the  velo- 
cities, while  all  other  circumstances  remained  the  same,  underwent 
successive  changes  from  a  walk  to  a  canter. 

Some  of  the  results  of  these  experiments  are  given  in  the  follow- 
ing table : — 


*  Empierrement  solide. 


f  Pav6  en  gr&s. 


310 


THE   PRACTICAL   MODEL   CALCULATOR. 


Carriage  employed. 

Ground  passed  over. 

Load. 

Pace. 

Rate  of 
speed, 
in  miles, 
per 
honr. 

Trac- 
tion. 

Ratio 
of  the 
traction 
to  the 
load. 

Apparatus  upon  a 

Ground  of  the  po- 

kii. 
1042 

Walk  

miles. 

3-13 

kil. 

•165-0 

1/6-32 

brass  shaft. 

lygon  at  Metz, 

Trot  

6-26 

168-0 

1/6-2 

•wet  and  soft. 

1335 

Walk  
Trot 

2-860 
7-560 

215-0 
197-0 

1/6-21 

1/6-78 

A  sixteen-pounder 

Road  from    Metz 

3750 

Walk 

2-820 

92- 

1/40-8 

carriage       and 
piece 

to       Montigny, 
solid        gravel 

*  Brisk  walk 
Trot 

3-400 
5-480 

92- 
102- 

1/40-8 
1/36-8 

Chariot  des  Mes- 

very  even   and 
very  dry. 
Pitched    road    of 

3288 

f  Canter  
Walk  

8-450 
2-770 

121- 
144- 

1/31- 

1/22-8 

sageries,      sus- 
pended upon  six 
springs. 

Fontainebleau. 

3353 

*Brisk  walk 
Trot  

3-82 

5-28 

153- 
161-   ' 

1/21-9 
1/20-8 

JBrisk  trot. 

8-05 

183-5 

1/18-3 

We  see,  by  these  examples,  that  the  traction  undergoes  no  sen- 
sible augmentation  with  the  increase  of  velocity  on  soft  grounds  ; 
but  that  on  solid  and  uneven  roads  it  increases  with  an  increase  of 
velocity,  and  in  a  greater  degree  as  the  ground  is  more  uneven,  and 
the  carriage  has  less  spring. 

To  find  the  relation  between  the  resistance  to  rolling  and  the  ve- 
locity, the  velocities  were  set  off  as  abscissas,  and  the  values  of  A 
furnished  by  the  experiments,  as  ordinates ;  and  the  points  thus 
determined  were,  for  each  series  of  experiments,  situated  very 
nearly  upon  a  straight  line.  The  value  of  A,  then,  can  be  repre- 
sented by  the  expression, 

A  =  a  -f  d  (V-  -  2) 

in  which  a  is  a  number  constant  for  each  particular  state  of  each 
kind  of  ground,  and  which  expresses  the  value  of  the  number  A  for 
the  velocity,  V  =  2  miles,  (per  hour,)  which  is  that  of  a  very  slow 
walk. 

d,  a  factor  constant  for  each  kind  of  ground  and  each  sort  of 
carriage. 

The  results  of  experiments  made  with  a  carriage  of  a  siege  train, 
with  its  piece,  gave,  on  the  Montigny  road,  §very  good  solid  gravel, — 

A  =  0-03215  x  0-00295  (V  -  2). 

On  the  ||pitched  road  of  Metz,  A  =  0-01936  x  0-08200  (V  -  2). 

These  examples  are  sufficient  to  show — 

1st.  That,  at  a  walk,  the  resistance  on  a  good  pitched  road  is 
less  than  that  on  very  good  solid  gravel,  very  dry. 

2d.  That,  at  high  speeds,  the  resistance  on  the  pitched  road  in- 
creases very  rapidly  with  the  velocity. 

On  rough  roads  the  resistance  increases  with  the  velocity  much 
more  slowly,  however,  for  carriages  with  springs. 


*  Pas  allonge*.  f  Grand  trot.  J  Trot  allonge*. 

§  En  tres  bon  empierrement.  ||  Pav6  en  gres  de  Sieack. 


STRENGTH   OF   MATERIALS.  311 

Thus,  for  a  chariot  des  Messageries  Generates,  on  a  pitched  road, 
the  experiments  gave  A  =  0-0117  X  0-00361  (V  —  2) ;  while,  with 
the  springs  wedged  so  as  to  prevent  their  action,  the  experiments 
gave,  for  the  same  carriage,  on  a  similar  road,  A  =  0*02723  X 
0-01312  (V  —  2).  At  a  speed  of  nine  miles  per  hour,  the  springs 
diminish  the  resistance  by  one-half. 

The  experiments  further  showed  that,  while  the  pitched  road  was 
inferior  to  a  *solid  gravel  road  when  dry  and  in  good  repair,  the 
latter  lost  its  superiority  when  muddy  or  out  of  repair. 

INFLUENCE  OF  THE  INCLINATION  OF  THE  TRACES. 

The  inclination  of  the  traces,  to  produce  the  maximum  effect,  is 
given  by  the  expression — 

ljf      AxO-96/r' 
hJ~---  r- 0-4./V 

in  which  h  =  the  height  of  the  fore  extremity  of  the  trace  above 
the  point  where  it  is  attached  to  the  carriage ;  b  =  the  horizontal 
distance  between  these  two  points.  /  is  the  radius  of  interior  of 
the  boxes,  and  r  the  radius  of  the  wheel. 

The  inclination  given  by  this  expression  for  ordinary  carriages 
is  very  small ;  and  for  trucks  with  wheels  of  small  diameter  it  is 
much  less  than  the  construction  generally  permits. 

It  follows,  from  the  preceding  remarks,  that  it  is  advantageous 
to  employ,  for  all  carriages,  wheels  of  as  large  a  diameter  as  can 
be  used,  without  interfering  with  the  other  essentials  to  the  pur- 
poses to  which  they  are  to  be  adapted.  Carts  have,  in  this  respect, 
the  advantage  over  wagons  ;  but,  on  the  other  hand,  on  rough  roads, 
the  thill  horse,  jerked  about  by  the  shafts,  is  soon  fatigued.  Now, 
by  bringing  the  hind  wheels  as  far  forward  as  possible,. and  placing 
the  load  nearly  over  them,  the  wagon  is,  in  effect,  transformed  into 
a  cart ;  only  care  must  be  taken  to  place  the  centre  of  gravity  of 
the  load  so  far  in  front  of  the  hind  wheels  that  the  wagon  may  not 
turn  over  in  going  up  hill. 

ON  THE  DESTRUCTIVE  EFFECTS   PRODUCED  BY  CARRIAGES  ON  THE  ROADS. 

If  we  take  stones  of  mean  diameter  from  2f  to  3J  inches,  and, 
on  a  road  slightly  moist  and  soft,  place  them  first  under  the  small 
wheels  of  a  diligence,  and  then  under  the  large  wheels,  we  find  that, 
in  the  former  case,  the  stones,  pushed  forward  by  the  small  wheels, 
penetrate  the  surface,  ploughing  and  tearing  it  up ;  while  in  the 
latter,  being  merely  pressed  and  leant  upon  by  the  large  wheels, 
they  undergo  no  displacement.* 

From  this  simple  experiment  we  are  enabled  to  conclude  that 
the  wear  of  the  roads  by  the  wheels  of  carriages  is  greater  the 
smaller  the  diameter  of  the  wheels. 

Experiments  having  proved  that  on  hard  grounds  the  traction 
was  but  slightly  increased  when  the  breadths  of  the  wheels  was 

*  En  empierrement. 


312  THE    PRACTICAL    MODEL    CALCULATOR. 

diminished,  we  might  also  conclude  that  the  wear  of  the  road  would 
be  but  slightly  increased  by  diminishing  the  width  of  the  felloes. 

Lastly,  the  resistance  to  rolling  increasing  with  the  velocity,  it 
was  natural  to  think  that  carriages  going  at  a  trot  would  do  more 
injury  to  the  roads  than  those  going  at  a  walk.  But  springs,  by 
diminishing  the  intensity  of  the  impacts,  are  able  to  compensate, 
in  certain  proportions,  for  the  effects  of  the  velocity. 

Experiments,  made  upon  a  grand  scale,  and  having  for  their 
object  to  observe  directly  the  destructive  effects  of  carriages  upon 
the  roads,  have  confirmed  these  conclusions. 

These  experiments  showed  that  with  equal  loads,  on  a  solid  gra- 
vel road,  wheels  of  two  inches  breadth  produced  considerably  more 
wear  than  those  of  4J  inches,  but  that  beyond  the  latter  width  there 
was  scarcely  any  advantage,  so  far  as  the  preservation  of  the  road 
was  concerned,  in  increasing  the  size  of  the  tire  of  the  wheel. 

Experiments  made  with  wheels  of  the  same  breadth,  and  of  dia- 
meters of  2-86  ft.,  4-77  ft.,  and  6-69  ft.,  showed  that  after  the 
carriage  of  10018*2  tons,  over  tracks  218*72  yards  long,  the  track 
passed  over  by  the  carriage  with  the  smallest  wheels  was  by  far 
the  most  worn ;  while,  on  that  passed  over  by  the  carriage  with 
the  wheels  of  6*69  ft.  diameter,  the  wear  was  scarcely  perceptible. 

Experiments  made  upon  two  wagons  exactly  similar  in  all  other 
respects,  but  one  with  and  one  without  springs,  showed  that  the 
wear  of  the  roads,  as  well  as  the  increase  of  traction,  after  the 
passage  of  4577*36  tons  over  the  same  track,  was  sensibly  the  same 
for  the  carriage  without  springs,  going  at  a  walk  of  from  2*237  to 
2*684  miles  per  hour,  and  for  that. with  springs,  going  at  a  trot  of 
from  7*158  to  8*053  miles  per  hour. 


HYDRAULICS. 

THE   DISCHARGE   OP  WATER   BY   SIMPLE   ORIFICES   AND   TUBES. 

THE  formulas  for  finding  the  quantities  of  water  discharged  in  a 
given  time  are  of  an  extensive  and  complicated  nature.  The  more 
important  and  practical  results  are  given  in  the  following  Deduc- 
tions. 

When  an  aperture  is  made  in  the  bottom  or  side  of  a  vessel  con- 
taining water  or  other  homogeneous  fluid,  the  whole  of  the  particles 
of  fluid  in  the  vessel  will  descend  in  lines  nearly  vertical,  until  they 
arrive  within  three  or  four  inches  of  the  place  of  discharge,  when 
they  will  acquire  a  direction  more  or  less  oblique,  and  flow  directly 
towards  the  orifice.  , 

The  particles,  however,  that  are  immediately  over  the  orifice,  de- 
scend vertically  through  the  whole  distance,  while  those  nearer  to 
the  sides  of  the  vessel,  diverted  into  a  direction  more  or  less  oblique 
as  they  approach  the  orifice,  move  with  a  less  velocity  than  the 
former ;  and  thus  it  is  that  there  is  produced  a  contraction  in  the 
size  of  the  stream  immediately  beyond  the  opening,  designated  the 
vena  contracta,  and  bearing  a  proportion  to  that  of  the  orifice  of 


HYDRAULICS.  313 

about  5  to  8,  if  it  pass  through  a  thin  plate,  or  of  6  to  8,  if  through 
a  short  cylindrical  tube.  But  if  the  tube  be  conical  to  a  length 
equal  to  half  its  larger  diameter,  having  the  issuing  diameter  less 
than  the  entering  diameter  in  the  proportion  of  26  to  33,  the  stream 
does  not  become  contracted. 

If  the  vessel  be  kept  constantly  full,  there  will  flow  from  the 
aperture  twice  the  quantity  that  the  vessel  is  capable  of  contain- 
ing, in  the  same  time  in  which  it  would  have  emptied  itself  if  not 
kept  supplied. 

1.  How  many  horse-power  (H.  P.)  is  required  to  raise  6000 
cubic  feet-  of  water  the  hour  from  a  depth  of  300  feet  ? 

A  cubic  foot  of  water  weighs  62 '5  Ibs.  Avoirdupois. 

6000  x  62-5 

gO —    -  =  6250,  the  weight  of  water  raised  a  minute. 

6250  x  300  =  1875000,  the  units  of  work  each  minute. 

1875000 
Then    QQQQQ    =  56-818  =  the  horse-power  required. 

2.  "What  quantity  of  water  may  be  discharged  through  a  cylin- 
drical mouth-piece  2  inches  in  diameter,  under  a  head  of  25  feet  ? 

2         1 
jo  =  "g~  of  a  foot;  .'.  the   area  of  the    cross    section  of  the 

mouth-piece,  in  feet,  is  g  x  g  x  -7854  =  -021816. 


Theory  gives  -021816  v/2  g  X  25  the  cubic  feet  discharged  each 
second ;  but  experiments  show  that  the  effective  discharge  is  97  per 
cent,  of  this  theoretical  quantity :  g  =  32-2. 

Hence,  -97  x  -021816  >/64-4  x  25  =  -84912,  the  cubic  feet 
discharged  each  second. 

•84912  x  62-5  =  53-0688  Ibs.  of  water  discharged  each  second. 

Effluent  water  produces,  by  its  vis  viva,  about  6  per  cent,  less  me- 
chanical effect  than  does  its  weight  by  falling  Jrom  the  height  of 
the  head. 

3.  What  quantity  of  water  flows  through  a  circular  orifice  in  a 
thin  horizontal  plate,  3  inches  in  diameter,  under  a  head  of  49  feet  ? 

Taking  the  contraction  of  the  fluid  vein  into  account,  the  velo- 
city of  the  discharge  is  about  97  per  cent,  of  that  given  by  theory. 

The  theoretic  velocity  is  v/2#  x  49  =  7  N/6-44  =  56-21. 
•97  X  56-21  =  54-523  =  the  velocity  of  the  discharge. 
The  area  of  the  transverse  section  of  the  contracted  vein  is  -64 
of  the  transverse  section  of  the  orifice. 

3        1 

—  =  -  =  -25,  and  (-25)2  X  -7854  =  -0490875  =  area  of  orifice. 

.-.  -64  x  -0490875  =  -031416,  the  area  of  the  transverse  section 
of  the  contracted  vein. 
2B 


314  THE    PRACTICAL    MODEL    CALCULATOR. 

X 

Hence,  54-523  x  -031416  =  1-7129,  the  cubic  feet  of  water 
discharged  each  second.  The  later  experiments  of  Poncelet, 
Bidone,  and  Lesbros  give  *563  for  the  coefficient  of  contraction. 
Water  issuing  through  lesser  orifices  give  greater  coefficients  of 
contraction,  and  become  greater  for  elongated  rectangles,  than  for 
those  which  approach  the  form  of  a  square. 

Observations  show  that  the  result  above  obtained  is  too  great ; 
•fs  of  this  result  are  found  to  be  very  near  the  truth. 

4  of  1-7129  =  1-0541. 
lo 

4.  What  quantity  of  water  flows  through  a  rectangular  aperture 
7*87  inches  broad,  and  3*94  inches  deep,  the  surface  of  the  water 
being  5  feet  above  the  upper  edge ;  the  plate  through  which  the 
water  flows  being  -125  of  an  inch  thick. 

•          7-87 

-Yg-  =  -65583,  decimal  of  a  foot. 

3-94 

-.n,--  =  -32833,  decimal  of  a  foot. 

5-  and  5-32833  are  the  heads  of  water  above  the  uppermost  and 
lowest  horizontal  surfaces. 

The  theoretical  discharge  will  be 

|  X  -65583  x/2y((5-328)1  -  (5)*)  =  3-9268  cubic  feet. 

Table  I.  gives  the  coefficient  of  efflux  in  this  case,  -615,  which 
is  found  opposite  5  feet  and  under  4  inches ;  for  3-94  is  nearly 
equal  4. 

3-9268  X  -615  =  2-415  cubic  feet,  the  effective  discharge. 

5.  What  water  is  discharged  through  a  rectangular  orifice  in  a 
thin  plate  6  inches  broad,  3  inches  deep,  under  a  head  of  9  feet 
measured  directly  over  the  orifice  ? 

6 

Vo  =  *5,  decimal  of  a  foot. 

3 
jo  =  "25,  decimal  of  a  foot. 

The  theoretical  discharge  will  be 

|  X  -5  V2~g  { (9-25)*  -  (9)* }  =  3-033  cubic  feet. 

Table  II.  gives  the  coefficient  of  efflux  between  '604  and  -606 ; 
we  shall  take  it  at  -605,  then 

3-03?  X  -605  =  1-833  cubic  feet,  the  effective  discharge. 

6.  A  weir  -82  feet  br.oad,  and  4-92  feet  head  of  water,  how  many 
cubic  feet  are  discharged  each  second  ? 

The  quantity  will  be 

c  X  -82  x/2<?(4-92)3;  g  =  32-2; 


HYDRAULICS. 


315 


TABLE  I. —  The  Coefficients  for  the  Efflux  through  rectangular  ori- 
fices in  a  thin  vertical  plate,  the  heads  are  measured  where  the 
water  may  be  considered  still. 


Head  of  water,  or 
distance  of  the 
surface  of  the 
water  from  the 
Zer    side   of 
orifice    in 
feet. 

HEIGHT  OF  ORIFICE. 

81-' 

In. 

4- 

In. 

2- 

In. 
1- 

In. 

•8 

In. 

•4 

•1 

•579 

•599 

•619 

•634 

•656 

•686 

•2 

•582 

•601 

•620 

•638 

•654 

•981 

•3 

•585 

•603 

•621 

•640 

•653 

•676 

•4 

•588 

•605 

•622 

•639 

•652 

•671 

•5 

•591 

•607 

•623 

•637 

•650 

•666 

•6 

•594 

•609 

•624 

•635 

•649 

•662 

•7 

•596 

•611 

•625 

•634 

•648 

•659 

•8 

•597 

•613 

•623 

•632 

•647 

•656 

•9 

•598 

•615 

•627 

•631 

•645 

•653 

1-0 

•599 

•616 

•628 

•630 

•644 

•650 

2-0 

•600 

•617 

•628 

•628 

•641 

•647 

3-0 

•601 

•617 

•626 

•626 

•638 

•644 

4-0 

•602 

•616 

•624 

•623 

•634 

•640 

5-0 

•604 

•615 

•621 

•621 

•630 

•635 

6-0 

•603 

•613 

•618 

•618 

•625 

•630 

7-0 

•602 

•611 

•615 

•615 

•621 

•625 

8-0 

•601 

•609 

•612 

•613 

•617 

•619 

9-0 

•600 

•606 

•609 

•610 

•614 

•613 

10-0 

•600 

•604 

•606 

•608 

•611 

•609 

TABLE  II. — The  Coefficients  for  the  Efflux  through  rectangular  ori- 
fices in  a  thin  vertical  plate,  the  heads  of  water  being  measured 
directly  over  the  orifice. 


Head  of  water,  or 
distance  of  the 
surface  of  the 
water  from  the 

EIGHT  OF  ORIFICE. 

tipper   side  of 
the   orifice    in 

lu. 

In. 

In. 

In. 

In. 

In. 

feet. 

8- 

4- 

2- 

!• 

•8 

4 

•1 

•593 

•613 

•637 

•659 

•685 

•708 

•2 

•593 

•612 

•636 

•656 

•680 

•701 

•3 

•593 

•613 

•635 

•653 

•676 

•694 

•4 

•594 

•614 

•634 

•650 

•672 

•687 

•5 

•595 

•614 

•633 

•647 

•668 

•681 

•6 

•597 

•615 

•632 

•644 

•664 

•675 

•7 

•598 

•615 

•631 

•641 

•660 

•669 

•8 

•599 

•616 

•630 

•638 

•655 

•663 

•9 

•601 

•616 

•629 

•635 

•650 

•657 

1-0 

•603 

•617 

•629 

•632 

•644 

•651 

2-0 

•604 

•617 

•626 

•628 

•640 

•646 

3-0 

•605 

•616 

•622 

•627 

•636 

•641 

4-0 

•604 

•614 

•618 

•624 

•632 

•636 

5-0 

•604 

•613 

•616 

•621 

•628 

•631 

6-0 

•603 

•612 

•613 

•618 

•624 

•626 

7-0 

•603 

•610 

•611 

•616 

•620 

•621 

8-0 

•602 

•608 

•609 

•614 

•616 

•617 

9-0 

•601 

•607 

•607 

•612 

•613 

•613 

10-0 

•601 

•603 

•606 

•610 

•610 

•609 

316 


THE  PRACTICAL  MODEL  CALCULATOR. 


c  is  termed  the  coefficient  of  efflux,  and  on  an  average  may  be  taken 
at  -4.     It  is  found  to  vary  from  -B85  to  -444. 

Then  -4  X  -82  ^(64-4)  (4-92p  =  2-670033,  the  cubic  feet  dis- 
charged each  second. 

7.  What  breadth  must  be  given  to  a  notch,  in  a  thin  plate,  with 
a  head  of  water  of  9  inches,  to  allow  10  cubic  feet  to  flow  each 
second  'f 

The  breadth  will  be  represented  by 

10  10 

,  ^  =  -        — =====  =  4-7963  feet. 

c  VZg  x  (-75)3       -4  x  x/64-4  x  (-75)3 

Changes  in  the  coefficients  of  efflux  through  convergent  sides 
often  present  themselves  in  practice  :  they  occur  in  dams  which  are 
inclined  to  the  horizon. 

Poncelet  found  the  coefficient  -8,  when  the  board  was  inclined 
45°,  and  the  coefficient  *74  for  an  inclination  of  63°  34',  that  is 
for  a  slope  of  1  for  a  base,  and  2  for  a  perpendicular. 

8.  If  a  sluice  board,  inclined  at  an  angle  of  50°,  which  goes 
across  a  channel  2-25  feet  broad,  is  drawn  out  -5  feet,  what  quan- 
tity of  water  will  be  discharged,  the  surface  of  the  water  standing 
4-  feet  above  the  surface  of  the  channel,  and  the  coefficient  of  efflux 
taken  at  -78  ? 

The  height  of  the  aperture  =  -5  sin.  50°  =  -3830222 ;  4-  and 
4-  -  -3830222  =  3-6169778,  are  the  heads  of  water. 

.-.  |x  2-25  x  -78  x  v/%  |  (4)*  -  (3-617)*  j  =  10-5257  cu- 
bic feet,  the  quantity  discharged. 

The  calculations  just  made  appertain  to  those  cases  where  th.e 
water  flows  from  all  sides  towards  the  aperture,  and  forms  a  con- 
tracted vein  on  every  side.  We  shall  next  calculate  in  cases  where 
the  water  flows  from  one  or  more  sides  to  the  aperture,  and  hence 
produces  a  stream  only 
partially  contracted,  m, 
n,  o,  p,  are  four  orifices  in 
the  bottom  ABCD  of  a 
vessel ;  the  contraction  by 
efflux  through  the  orifice 
o,  in  the  middle  of  the  bot- 
tom, is  general,  as  the  water 
can  flow  to  it  from  all 
sides ;  the  contraction 
from  the  efflux  through  m,  n,  p,  is  partial,  as  the  water  can  only 
flow  to  them  from  one,  two,  or  three  sides.  Partial  contraction 
gives  an  oblique  direction  to  the  stream,  and  increases  the  quantity 
discharged. 

9.  What  quantity  of  water  is  delivered  through  a  flow  4  feet 
broad,  and  1  foot  deep,  vertical  aperture,  at  a  pressure  of  2  feet 
above  the  upper  edge,  supposing  the  lower  edge  to  coincide  with 


in 

1 

I 

— 

n 

\ 

\ 

•*••     o 

\ 

p  •* 

HYDRAULICS. 


317 


the  lower  side  of  the  channel,  so  that  there  is  no  contraction  at  the 
bottom  ? 

The  theoretical  discharge  will  be 

|  x  |  x  v/2<7  |  (3)1  -  (2)f  |  =  50-668  cubic  feet. 

The  coefficient  of  contraction  given  in  the  table  page  315,  may 
be  taken  at  '603. 

I. — Comparison  of  the,  Theoretical  with  the  Heal  Discharges  from  an 

Orifice. 


Constant  height 
of  the  water  in  the 
reservoir  above 
the  centre  of  the 
orifice. 

Theoretical  dis- 
charge through  a 
circular  orifice 
one  inch  in  di- 
ameter. 

Eeal  discharge 
in  the  same  time 
through  the  same 
orifice. 

Ratio  of  the 
theoretical  to  the  real 
discharge. 

Paris  Feet. 

Cubic  Inches. 

Cubic  Inches. 

1 

4381 

2722 

to  0-62133 

2 

6196 

3846 

to  0-62073 

3 

7589 

4710 

to  0-62064 

4 

8763 

5436 

to  0-62034 

5 

9797 

6075 

to  0-62010 

6 

10732 

6654 

to  0-62000 

7 

11592 

7183 

to  0-61965 

8 

12392 

7672 

to  0-61911 

9 

13144 

8135 

to  0-61892 

10 

13855 

8574 

to  0-61883 

11 

14530 

8990 

to  0-61873 

12 

15180 

9384 

to  0-61819 

13 

15797 

9764 

to  0-61810 

14 

16393 

10130 

to  0-61795 

15 

16968 

10472 

1  to  0-61716 

II. — Comparison  of  the  Theoretical  with  the  Heal  Discharges  from 

a  Tube. 


Constant  height 
of  the  water  in  the 
reservoir  above 
the  centre  of  the 
orifice. 

Theoretical  dis- 
charge through  a 
circular  orifice 
one  inch  in  di- 
ameter. 

Real  discharge 
in  the  same  time 
by  a  cylindrical 
tube  one  inch  in 
diameter  and  two 
inches  long. 

Ratio  of  the 
theoretical  to  the  real 
discharge. 

Paris  Feet. 

Cubic  Inches. 

Cubic  Inches. 

1 

4381 

3539 

1  to  0-81781 

2 

6196 

5002 

1  to  0-80729 

3 

7589 

6126 

1  to  0-80724 

4 

8763 

7070 

1  to  0-80681 

5 

9797 

7900 

1  to  0-80638 

6 

10732 

8654 

1  to  0-80638 

7 

11592 

9340 

1  to  0-80577 

8 

12392 

9975 

1  to  0-80496 

9 

13144 

10579 

1  to  0-80485  , 

10 

13855 

11151 

1  to  0-80483 

11 

14530 

11693 

1  to  0-80477 

12 

15180 

12205 

1  to  0-80403 

13 

15797 

12699 

1  to  0-80390 

14 

16393 

13177 

1  to  0-80382 

15 

16968 

13620 

1  to  0-80270 

2  B2 


318  THE   PRACTICAL   MODEL   CALCULATOR. 

THE    DISCHARGE    BY   DIFFERENT    APERTURES    AND    TUBES,    UNDER    DIF- 
FERENT   HEADS    OF   WATER. 

The  velocity  of  water  flowing  out  of  a  horizontal  aperture,  is  as 
the  square  root  of  the  height  of  the  head  of  the  water. — That  is,  the 
pressure,  and  consequently  the  height,  is  as  the  square  of  the  ve- 
locity ;  for,  the  quantity  flowing  out  in  any  short  time  is  as  the 
velocity ;  and'the  force  required  to  produce  a  velocity  in  a  certain 
quantity  of  matter  in  a  given  time  is  also  as  that  velocity ;  there- 
fore, the  force  must  be  as  the  square  of  the  velocity. 

Or,  supposing  a  very  small  cylindrical  plate  of  water,  imme- 
diately over  the  orifice,  to  be  put  in  motion  at  each  instant,  by  the 
pressure  of  the  whole  cylinder  upon  it,  employed  only  in  generat- 
ing its  velocity;  this  plate  would  be  urged  by  a  force  as  much 
greater  than  its  own  weight  as  the  column  is  higher  than  itself, 
through  a  space  shorter  in  the  same  proportion  than  that  height. 
But  where  the  forces  are  inversely  as  the  spaces  described,  the 
final  velocities  are  equal.  Therefore,  the  velocity  of  the  water 
flowing  out  must  be  equal  to  that  of  a  heavy  body  falling  from  the 
height  of  the  head  of  water ;  which  is  found,  very  nearly,  by  mul- 
tiplying the  square  root  of  that  height  in  feet  by  8,  for  the  number 
of  feet  described  in  a  second.  Thus,  a  head  of  1  foot  gives  8 ;  a 
head  of  9  feet,  24.  This  is  the  theoretical  velocity ;  but,  in  con- 
sequence of  the  contraction  of  the  stream,  we  must,  in  order  to  ob- 
tain the  actual  velocity,  multiply  the  square  root  of  the  height,  in 
feet,  by  5  instead  of  8. 

The  velocity  of  a  fluid  issuing  from  an  aperture  is  not  affected 
by  its  density  being  greater  or  less.  Mercury  and  water  issue 
with  equal  velocities  at  equal  altitudes. 

The  proportion  of  the  theoretical  to  the  actual  velocity  of  a  fluid 
issuing  through  an  opening  in  a  thin  substance,  according  to  M. 
Eytelwein,  is  as  1  to  -619 ;  but  more  recent  experiments  make  it 
as  1  to  -621  up  to  -645. 

APPLICATION  OF  THE  TABLES  IN  THE  PRECEDING  PAGE. 

TABLE  I. —  To  find  the  quantities  of  water  discharged  ly  orifices 
of  different  sizes  under  different  altitudes  of  the  fluid  in  the  reser- 
voir. 

To  find  the  quantity  of  fluid  discharged  by  a  circular  aperture 
3  inches  in  diameter,  the  constant  altitude  being  30  feet. 

As  the  real  discharges  are  in  the  compound  ratio  of  the  area  of 
the  apertures  and  the  square  roots  of  the  altitudes  of  the  water, 
and  as  the  theoretical  quantity  of  water  discharged  by  an  orifice 
one  inch  in  diameter  from  a  height  of  15  feet  is,  by  the  second  co- 
lumn of  the  table,  16968  cubic  inches  in  a  minute,  we  have  this 
proportion :  1  </15  :  9  v/30  : :  16968  :  215961  cubic  inches ;  the 
theoretical  quantity  required.  This  quantity  being  diminished  in 
the  ratio  of  1  to  -62,  being  the  ratio  of  the  theoretical  to  the  ac- 
tual discharge,  according  to  the  fourth  column  of  the  table,  gives 
133896  cubic  inches  for  the  actual  quantity  of  water  discharged  by 


HYDRAULICS.  319 

the  given  aperture.  Hence,  the  quantity  should  be  rather  greater, 
because  large  orifices  discharge  more  in  proportion  than  small  ones ; 
while  it  should  be  rather  less,  because  the  altitude  of  the  fluid 
being  greater  than  that  in  the  table  with  which  it  is  compared,  the 
flowing  vein  of  water  becomes  rather  more  contracted.  The  quan- 
tity thus  found,  therefore,  is  nearly  accurate  as  an  average. 

When  the  orifice  and  altitude  are  less  than  those  in  the  table,  a 
few  cubic  inches  should  be  deducted  from  the  result  thus  derived. 

The  altitude  of  the  fluid  being  multiplied  by  the  coefficient  8-016 
will  give  its  theoretical  velocity ;  and  as  the  velocities  are  as  the 
quantities  discharged,  the  real  velocity  may  be  deducted  from  the 
theoretical  by  means  of  the  foregoing  results. 

TABLE  II. —  To  find  the  quantities  of  ivater  discharged  by  tubes 
of  different  diameter,  and  under  different  heights  of  water. 

To  find  the  quantity  of  water  discharged  by  a  cylindrical  tube, 
4  inches  in  diameter,  and  8  inches  long,  the  constant  altitude  of 
the  water  in  the  reservoir  being  25  feet. 

Find,  in  the  same  manner  as  by  the  example  to  Table  I.,  the 
theoretical  quantity  discharged,  which  is  furnished  by  this  analogy. 
1  >/15  :  16  v/25  : :  16968  :  350490  cubic  inches,  the  theoretical 
discharge.  This,  diminished  in  the  ratio  of  1  to  '81  by  the  4th 
column,  will  give  28473  cubic  inches  for  the  actual  quantity  dis- 
charged. If  the  tube  be  shorter  than  twice  its  diameter,  the 
quantity  discharged  will  be  diminished,  and  approximate  to  that 
from  a  simple  orifice,  as  shown  by  the  production  of  the  vena  con- 
tracta  already  described. 

According  to  Eytelwein,  the  proportion  of  the  theoretical  to  the 
real  discharge  through  tubes,  is  as  follows  : 

Through  the  shortest  tube  that  will  cause  the  stream  to  adhere 
everywhere  to  its  sides,  as  1  to  0'8125. 

Through  short  tubes,  having  their  lengths  from  two  to  four 
times  their  diameters,  as  1  to  0'82. 

Through  a  tube  projecting  within  the  reservoir,  as  1  to  0'50. 

It  should,  however,  be  stated,  that  in  the  contraction  of  the 
stream  the  ratio  is  not  constant.  It  undergoes  perceptible  varia- 
tions by  altering  the  form  and  position  of  the  orifice,  the  thickness 
of  the  plate,  the  form  of  the  vessel,  and  the  velocity  of  the  issu- 
ing fluid. 

Deductions  from  experiments  made  by  Bossut,  Michelloti. 

1.  That  the  quantities  of  fluid  discharged  in  equal  times  from 
different-sized  apertures,   the  altitude  of  the  fluid  in   the  reser- 
voir being  the  same,  are  to  each  other  nearly  as  the  area  of  the  aper- 
tures. 

2.  That  the  quantities  of  water  discharged  in  equal  times  by 
the  same  orifice  under  different  heads  of  water,  are  nearly  as  the 
square  roots  of  the  corresponding  heights  of  water  in  the  reservoir 
above  the  centre  of  the  apertures. 


320  THE   PRACTICAL   MODEL   CALCULATOR. 

3.  That,  in  general,  the  quantities  of  water  discharged,  in  the 
same  time,  by  different  apertures  under  different  heights  of  water 
in  the  reservoir,  are  to  one  another  in  the  compound  ratio  of  the 
areas  of  the  apertures,  and  the  square  roots  of  the  altitudes  of  the 
water  in  the  reservoirs. 

4.  That  on  account  of  the  friction,  the  smallest  orifice  discharges 
proportionally  less  water  than  those  which  are  larger  and  of  a 
similar  figure,  under  the  same  heads  of  water. 

5.  That,  from  the  same  cause,  of  several  orifices  whose  areas 
are  equal,  that  which  has  the  smallest*  perimeter  will  discharge 
more  water  than  the  other,  under  the  same  altitudes  of  water  in 
the  reservoir.     Hence,  circular  apertures  are  most  advantageous,  as 
they  have  less  rubbing  surface  under  the  same  area. 

6.  That,  in  consequence  of  a  slight  augmentation  which  the 
contraction  of  the  fluid  vein  undergoes,  in  proportion  as  the  height 
of  the  fluid  in  the  reservoir  increases,  the  expenditure  ought  to  be 
a  little  diminished. 

7.  That  the  discharge  of  a  fluid  through  a  cylindrical  horizontal 
tube,  the  diameter  and  length  of  which  are  equal  to  one  another, 
is  the  same  as  through  a  simple  orifice. 

8.  That  if  the  cylindrical  horizontal  tube  be  of  greater  length 
than  the  extent  of  the  diameter,  the  discharge  of  water  is  much 
increased. 

9.  That  the  length  of  the  cylindrical  horizontal  tube  may  be 
increased  with  advantage  to  four  times  the  diameter  of  the  orifice. 

10.  That  the  diameters  of  the  apertures  and  altitudes  of  water 
in  the  reservoir  being  the  same,  the  theoretic  discharge  through  a 
thin  aperture,  which  is  supposed  to  have  no  contraction  in  the  vein, 
the  discharge  through  an  additional  cylindrical  tube  of  greater 
length  than  the  extent  of  its  diameter,  and  the  actual  discharge 
through  an  aperture  pierced  in  a  thin  substance,  are  to  each  other 
as  the  numbers  16,  13,  10. 

11.  That  the  discharges  by  different  additional  cylindrical  tubes, 
under  the  same  head  of  water,  are  nearly  proportional  to  the  areas 
of  the  orifices,  or  to  the  squares  of  the  diameters  of  the  orifices. 

12.  That  the  discharges  by  additional  cylindrical  tubes  of  the 
same  diameter,  under  different  heads  of  water,  are  nearly  propor- 
tional to  the  square  roots  of  the  head  of  water. 

13.  That  from  the  two  preceding  corollaries  it  follows,  in  gene- 
ral, that  the  discharge  during  the  same  time,  by  different  addi- 
tional tubes,  and  under  different  heads  of  water  in  the  reservoir, 
are  to  one  another  nearly  in  the  compound  ratio  of  the  squares 
of  the  diameters  of  the  tubes,  and  the  square  roots  of  the  heads 
of  water. 

The  discharge  of  fluids  by  additional  tubes  of  a  conical  figure, 
when  the  inner  to  the  outer  diameter  of  the  orifice  is  as  33  to 
26,  is  augmented  very  nearly  one-seventeenth  and  seven-tenths 
more  than  by  cylindrical  tubes,  if  the  enlargement  be  not  carried 
too  far. 


HYDRAULICS.  321 

DISCHARGE   BY   COMPOUND   TUBES. 

Deductions  from  the  experiments  of  M.   Venturi. 

In  the  discharge  by  compound  tubes,  if  the  part  of  the  addi- 
tional tube  nearest  the  reservoir  have  the  form  of  the  contracted 
vein,  the  expenditure  will  be  the  same  as  if  the  fluid  were  not  con- 
tracted at  all ;  and  if  to  the  smallest  diameter  of  this  cone  a  cylin- 
drical pipe  be  attached,  of  the  same  diameter  as  the  least  section 
of  the  contracted  vein,  the  discharge  of  the  fluid  will,  in  a  horizon- 
tal direction,  be  lessened  by  the  friction  of  the  water  against  the 
side  of  the  pipe ;  but  if  the  same  tube  be  applied  in  a  vertical 
direction,  the  expenditure  will  be  augmented,  on  the  principle  of 
the  gravitation  of  falling  bodies;  consequently,  the  greater  the 
length  of  pipe,  the  more  abundant  is  the  discharge  of  fluid. 

If  the  additional  compound  tube  have  a  cone  applied  to  the  op- 
posite extremity  of  the  pipe,  the  expenditure  will,  under  the  same 
head  of  water,  be  increased,  in  comparison  with  that  through  a 
simple  orifice,  in  the  ratio  of  24  to  10. 

In  order  to  produce  this  singular  effect,  the  cone  nearest  to  the 
reservoir  must  be  of  the  form  of  the  contracted  vein,  which  will 
increase  the  expenditure  in  the  ratio  of  12*1  to  10.  At  the  other 
extremity  of  the  pipe,  a  truncated  conical  tube  must  be  applied, 
of  which  the  length  must  be  nearly  nine  times  the  smaller  diameter, 
and  its  outward  diameter  must  be  1-8  times  the  smaller  one.  This 
additional  cone  will  increase  the  discharge  in  the  proportion  of 
24  to  10.  But  if  a  great  length  of  pipe  intervene,  this  additional 
tube  has  little  or  no  effect  on  the  quantity  discharged. 

According  to  M.  Venturi's  experiments  on  the  discharge  of 
water  by  bent  tubes,  it  appears  that  while,  with  a  height  of  water 
in  the  reservoir  of  32-5  inches,  4  Paris  cubic  feet  were  discharged 
through  a  cylindrical  horizontal  tube  in  the  space  of  45  seconds, 
the  discharge  of  the  same  quantity  through  a  tube  of  the  same 
diameter,  with  a  curved  end,  occupied  50  seconds,  and  through  a 
like  tube  bent  at  right  angles,  70  seconds.  Therefore,  in  making 
cocks  or  pipes  for  the  discharge  or  conveyance  of  water,  great 
attention  should  be  paid  to  the  nature  and  angle  of  the  bendings ; 
right  angles  should  be  studiously  avoided. 

The  interruption  of  the  discharge  by  various  enlargements  of 
the  diameter  of  the  tubes  having  been  investigated  by  M.  Venturi, 
by  means  of  a  tube  with  a  diameter  of  9  lines,  enlarged  in  several 
parts  to  a  diameter  of  24  lines,  the  retardation  was  found  to  in- 
crease nearly  in  proportion  to  the  number  of  enlargements ;  the 
motion  of  the  fluid,  in  passing  into  the  enlarged  parts,  being 
diverted  from  its  direct  course  into  eddies  against  the  sides  of  the 
enlargements.  From  which  it  may  be  deduced,  that  if  the  inter- 
nal roughness  of  a  pipe  diminish  the  expenditure,  the  friction  of 
the  water  against  these  asperities  does  not  form  any  considerable 
part  of  the  cause.  A  right-lined  tube  may  have  its  internal  sur- 
face highly  polished  throughout  its  whole  length,  and  it  may  every- 

21 


322 


THE  PRACTICAL  MODEL  CALCULATOR. 


where  possess  a  diameter  greater  than  the  orifice  to  which  it  is 
applied ;  but,  nevertheless,  the  expenditure  will  be  greatly  retarded 
if  the  pipe  should  have  enlarged  parts  or  swellings.  It  is  not 
enough  that  elbows  and  contractions  be  avoided ;  for  it  may  hap- 
pen, by  an  intermediate  enlargement,  that  the  whole  of  the  other 
advantage  may  be  lost.  This  will  be  obvious  from  the  results  in 
the  following  table,  deduced  from  experiments  with  tubes  having 
various  enlargements  of  diameter. 


Head  of  water 
in  inches. 

Number  of  en- 
larged parts. 

Seconds  in  which 
4  cubic  feet  were 
discharged. 

32-5 

0 

109 

32-5 

1 

MT 

32-5 

3 

192 

32-5 

5 

240   - 

DISCHARGE   BY   CONDUIT   PIPES. 

On  account  of  the  friction  against  the  sides,  the  less  the  dia- 
meter of  the  pipe,  the  less  proportionally  is  the  discharge  of  fluid. 
And,  from  the  same  cause,  the  greater  the  length  of  conduit  pipe, 
the  greater  the  diminution  of  the  discharge.  Hence,  the  dis- 
charges made  in  equal  times  by  horizontal  pipes  of  different  lengths, 
but  of  the  same  diameter,  and  under  the  same  altitude  of  water, 
are  to  one  another  in  the  inverse  ratio  of  the  square  roots  of  the 
lengths.  In  order  to  have  a  perceptible  and  continuous  discharge 
of  fluid,  the  altitude  of  the  water  in  the  reservoir,  above  the  axis 
of  the  conduit  pipe,  must  not  be  less  than  1-|  inch  for  every  180 
feet  of  the  length  of  the  pipe. 

The  ratio  of  the  difference  of  discharge  in  pipes,  16  and  24  lines 
diameter  respectively,  may  be  known  by  comparing  the  ratios  of 
Table  I.  with  the  ratios  of  Table  II.,  in  the  following  page. 

The  greater  the  angle  of  inclination  of  a  conduit  pipe,  the 
greater  will  be  the  discharge  in  a  given  time ;  but  when  the  angle 
of  the  conduit  pipe  is  6°  31',  or  the  depression  of  the  lower  extre- 
mity of  the  pipe  is  one-eighth  or  one-ninth  of  its  length,  the  rela- 
tive gravity  of  the  fluid  will  be  counterbalanced  by  the  resistance 
or  friction  against  the  sides ;  and  the  discharge  is  then  the  same 
as  by  an  additional  horizontal  tube  of  the  same  diameter. 

A  curvilinear  pipe,  the  altitude  of  the  water  in  the  reservoir  being 
the  same,  discharges  less  water  when  the  flexures  lie  horizontally, 
than  a  rectilinear  pipe  of  the  same  diameter  and  length. 

The  discharge  by  a  curvilinear  pipe  of  the  same  diameter  and 
length,  and  under  the  same  head  of  water,  is  still  further  dimi- 
nished when  the  flexures  lie  in  a  vertical  instead  of  a  horizontal  plane. 

When  there  is  a  number  of  contrary  flexures  in  a  large  pipe,  the 
air  sometimes  lodges  in  the  highest  parts  of  the  flexures,  and  greatly 
retards  the  motion  of  the  water,  unless  prevented  by  air-holes,  or 
stopcocks. 


HYDRAULICS. 


323 


TABLE  I. — Comparison  of  the  discharge  by  conduit  pipes  of  different 
lengths,  16  lines  in  diameter,  with  the  discharge  by  additional 
tubes  inserted  in  the  same  reservoir. — By  M.  BOSSUT.. 


Constant 

altitude  of  the 

Length  of 

Quantity  of  Water  discharged 
in  a  minute. 

Ratio  between  the 

Water  above  the 
centre  of  the 
aperture. 

the  conduit 
pipe. 

by  additional 
tube,  16  lines  in 
diameter. 

by  conduit 
pipe,  16  lines  in 
diameter. 

quantities  furnished 
by  tube  and  pipe. 

Feet. 

Feet. 

Cubic  Inches. 

Cubic  Inches. 

1        ' 

30 

6330 

2778 

100  to  43-39 

1 

60 

6330 

1957 

100  to  30-91 

1 

90 

6330 

1587 

100  to  25-07 

1 

120 

6330 

1351 

100  to  21  -34 

1 

150 

6330 

1178 

100  to  18-61 

1 

180 

6330 

1052 

100  to  16-62 

2 

30 

8939 

4066 

100  to  45-48 

2 

60 

8939 

2888 

100  to  32-31 

2 

90 

8939 

2352 

100  to  26-31 

2 

120 

8939 

2011 

100  to  22-50 

2 

150 

8939 

1762 

100  to  19-71 

2 

180 

8939 

1583 

100  to  17-70 

TABLE  II. — Comparison  of  the  discharge  by  conduit  pipes  of  dif- 
ferent lengths,  24  lines  in  diameter,  with  the  discharge  by  addi- 
tional tubes  inserted  in  the  same  reservoir. — Bj  M.  BOSSUT. 


Constant 
altitude  of  the 

Length  of 

Quantity  of  Water  discharged 
in  a  minute. 

Ratio  between  the 

Water  above  the 
centre  of  the 
aperture. 

the  conduit 
pipe. 

by  additional 
tube,  24  lines  in 
diameter. 

by  conduit 
pipe,  24  lines  in 
diameter. 

quantities  furnished 
by  tube  and  pipe. 

Feet. 

Feet. 

Cubic  Inches. 

Cubic  Inches. 

1 

30 

14243 

7680 

100  to  53-92 

1 

60 

14243 

5564 

100  to  39-06 

1 

90 

14243 

4534 

100  to  31  -83 

1 

120 

14243 

3944 

100  to  27-69 

1 

150 

14243 

3486 

100  to  24-48 

1 

180 

14243 

3119 

100  to  21  -90 

2 

30 

20112 

11219 

100  to  55-78 

2 

60 

20112 

8190 

100  to  40-72 

2 

90 

20112 

6812 

100  to  33-87 

2 

120 

20112 

5885 

100  to  29-26 

2 

150 

20112 

5232 

100  to  26-01 

2 

180 

20112 

4710 

100  to  23-41 

DISCHARGE   BY    WEIRS    AND   RECTANGULAR   APERTURES. 

Rectangular  orifices  in  the  side  of  a  reservoir,  extending  to  the  surface. 

The  velocity  varying  nearly  as  the  square  root  of  the  height, 
may  here  be  represented  by  the  ordinates  of  a  parabola,  and  the 
quantity  of  water  discharged  by  the  area  of  the  parabola,  or 
two-thirds  of  that  of  the  circumscribing  rectangle.  So  that  the 
quantity  discharged  may  be  found  by  taking  two-thirds  of  the  velo- 
city due  to  the  mean  height,  and  allowing  for  the  contraction  of 
the  stream,  according  to  the  form  of  the  opening. 

In  a  lake,  for  example,  in  the  side  of  which  a  rectangular  open- 
ing is  made  without  any  oblique  lateral  walls,  three  feet  wide,  and 


324  THE   PRACTICAL   MODEL   CALCULATOR. 

extending  two  feet  below  the  surface  of  the  water,  the  coefficient 
of  the  velocity,  corrected  for  contraction,  is  5-1,  and  the  corrected 
mean  velocity  f  >/2  x  5-1  =  4-8 ;  therefore  the  area  being  6,  the 
discharge  of  water  in  a  second  is  28 -8  cubic  feet,  or  nearly  four 
hogsheads. 

The  same  coefficient  serves  for  determining  the  discharge  over 
a  weir  of  considerable  breadth ;  and,  hence,  to  deduce  the  depth 
or  breadth  requisite  for  the  discharge  of  a  given  quantity  of  water. 
For  example,  a  lake  has  a  weir  three  feet  in  breadth,  and  the  sur- 
face of  the  water  stands  at  the  height  of  five  feet  above  it :  it  is 
required  how  much  the  weir  must  be  widened,  in  order  that  the 
water  may  be  a  foot  lower.  Here  the  velocity  is  f  v/5  X  5-1,  and  the 
quantity  of  water  f  >/5  X  5-1  X  3  x  5 ;  but  the  velocity  must  be  re- 

-r  |  v/5^x  5-1x3x5 

duced  to  f  %/4  X  5-1,  and  then  the  section  will  be  —     ==— 

f  >/4  x  5-1 
/P.  x  S  x  5 
=  -      — j =  7-5  X  x/5 ;  and  the  height  being  4,  the  breadth 

7-5 

must  be  -j-  \/5  =  4'19  feet. 

The  discharge  from  reservoirs,  with  lateral  orifices  of  consider- 
able magnitude,  and  a  constant  head  of  water,  may  be  found  by 
determining  the  difference  in  the  discharge  by  two  open  orifices  of 
different  heights ;  or,  in  most  cases,  with  nearly  equal  accuracy, 
by  considering  the  velocity  due  to  the  distance,  below  the  surface, 
of  the  centre  of  gravity  of  the  orifice. 

Under  the  same  height  of  water  in  the  reservoir,  the  same  quan- 
tity always  flows  in  a  canal,  of  whatever  length  and  declivity ;  but 
in  a  tube,  a  difference  in  length  and  declivity  has  a  great  effect  on 
the  quantity  of  water  discharged. 

The  velocity  of  water  flowing  in  a  river  or  stream  varies  at  dif- 
ferent parts  of  the  same  transverse  section.  It  is  found  to  be 
greatest  where  the  water  is  deepest,  at  somewhat  less  than  one- 
half  the  depth  from  the  surface ;  diminishing  towards  the  sides 
and  shallow  parts. 

Resistance  to  bodies  moving  in  fluids. — The  deductions  from  the 
experiments  of  C.  Colles,  (who  first  planned  the  Croton  Aqueduct, 
New  York,)  and  others,  on  this  intricate  subject,  are,  as  stated,  thus : 

~L  The  confirmation  of  the  theory,  that  the  resistance  of  fluids 
to  passing  bodies  is  as  the  squares  of  the  velocities. 

2.  That,   contrary  to  the  received  opinion,   a  cone  will  move 
through  the  water  with  much  less  resistance  with  its  apex  foremost, 
than  with  its  base  forward. 

3.  That  the  increasing  the  length  of  a  solid,  of  almost  any  form, 
by  the  addition  of  a  cylinder  in  the  middle,  diminishes  the  resist- 
ance with  which  it  moves,  provided  the  weight  in  the  water  remains 
the  same. 


HYDRAULICS. 


325 


4.  That  the  greatest  breadth  of  the  moving  body  should  be 
placed  at  the  distance  of  two-fifths  of  the  whole  length  from  the 
bow,  when  applied  to  the  ordinary  forms  in  naval  architecture. 

5.  That  the  bottom  of  a  floating  solid  should  be  made  triangu- 
lar ;  as  in  that  case  it  will  meet  with  the  least  resistance  when 
moving  in  the  direction  of  its  longest  axis,  and  with  the  greatest 
resistance  when  moving  with  its  broadside  foremost. 

Friction  of  fluids. — Some  experiments  have  been  made  on  this 
subject,  with  reference  to  the  motion  of  bodies  in  water,  upon  a 
cylindrical  model,  30  inches  in  length,  26  inches  in  diameter,  and 
weighing  255  Ibs.  avoirdupois.  The  cylinder  was  placed  in  a  cis- 
tern of  salt  water,  and  made  to  vibrate  on  knife-edges  passing 
through  its  axis,  and  was  deflected  over  to  various  angles  by  means 
of  a  weight  attached  to  the  arm  of  a  lever.  The  experiments  were 
then  repeated  without  the  water,  and  the  following  are  the  angles 
of  deflection  and  vibration  in  the  two  cases. 


In  the  salt  water. 

In  the  atmosphere. 

Angle  of 
Deflection. 

Angle  to  which 
it  vibrated. 

Angle  of 
Deflection. 

Angle  to  which 
it  vibrated. 

22°  30' 

22°  24' 

22°  30' 

20°  V 

22   10 

22     6 

21   36 

21    3 

21   54 

21    48 

20  48 

2016 

21    36 

21    30 

&c. 

&c. 

&c. 

&c. 

Showing  that  the  amplitude  of  vibration  when  oscillating  in  water 
is  considerably  less  than  when  oscillating  without  water.  In  the 
experiments  there  is  a  falling  off  in  the  angle  of  24',  or  nearly 
half  a  degree.  The  amount  of  force  acting  on  the  surface  of  the 
cylinder  necessary  to  cause  the  above  difference  was  calculated ; 
and  the  author  thinks  that  it  is  not  equally  distributed  on  the 
surface  of  the  cylinder,  but  that  the  amount  on  any  particular 
part  might  vary  as  the  depth.  On  this  supposition,  a  constant 
pressure  at  a  unit  of  depth  is  assumed,  and  this,  multiplied  by  the 
depth  of  any  other  point  of  the  cylinder  immersed  in  the  water, 
will  give  the  pressure  at  that  point.  These  forces  or  moments 
being  summed  by  integration  and  equated  with  the  sum  of  the 
moments  given  by  the  experiments,  we  have  the  value  of  the  con- 
stant pressure  at  a  unit  of  depth  =  -0000469.  This  constant,  in 
another  experiment,  the  weight  of  the  model  being  197  Ibs.  avoir- 
dupois, and  consequently  the  part  immersed  in  the  water  being  dif- 
ferent from  that  in  the  other  experiment,  was  '0000452,  which 
differs  very  little  from  the  former, — indicating  the  probability  of 
the  correctness  of  the  assumption. 

The  drainage  of  water  through  pipes. — The  experiments  made 

under  the  direction  of  the  Metropolitan  Commissioners  of  Sewers, 

on  the  capacities  of  pipes  for  the  drainage  of  towns,  have  presented 

some  useful  results  for  the  guidance  of  those  who  have  to  make 

2C 


326 


THE  PRACTICAL  MODEL  CALCULATOR. 


calculations  for  a  similar  purpose.  The  pipes,  of  various  dia- 
meters, from  3  to  12  inches,  were  laid  on  a  platform  of  100  feet 
in  length,  the  declivity  of  which  could  be  varied  from  a  horizontal 
level  to  a  fall  of  1  in  10.  The  water  was  admitted  at  the  head 
of  the  pipe,  and  at  five  junctions,  or  tributary  pipes  on  each  side, 
so  regulated  as  to  keep  the  main  pipe  full. 

The  results  were  as  follow  : — 

It  was  found — to  mention  only  one  result — that  a  line  of  6-inch 
pipes,  100  feet  long,  at  an  inclination  of  1  in  60,  discharged  75  cubic 
feet  per  minute.  The  same  experiment,  repeated  with  the  line  of 
pipes  reduced  to  50  feet  in  length,  gave  very  nearly  the  same  result. 
Without  the  addition  of  junctions,  the  transverse  sectional  area  of 
the  stream  of  water  near  the  discharging  end  was  reduced  to  one- 
fifth  of  the  corresponding  area  of  the  pipe,  and  it  required  a  sim- 
ple head  of  water  of  about  22  inches  to  give  the  same  result  as 
that  accruing  under  the  circumstances  of  the  junctions.  With 
regard  to  varying  sizes  and  inclinations,  it  appears,  sufficiently  for 
practical  purposes,  that  the  squares  of  the  discharges  are  as  the 
fifth  powers  of  the  diameters ;  and  again,  that  in  steeper  declivi- 
ties than  1  in  70,  the  discharges  are  as  the  square  roots  of  the 
inclinations ;  but  at  less  declivities  than  1  in  70,  the  ratios  of  the 
discharges  diminish  very  rapidly,  ,and  are  governed  by  no  constant 
law.  At  a  certain  small  declivity,  the  relative  discharge  is  as  the 
fifth  root  of  the  inclination ;  at  a  smaller  declivity,  it  is  found  as 
the  seventh  root  of  the  inclination ;  and  so  on,  as  it  approaches  the 
horizontal  plane.  This  may  be  exemplified  by  the  following  results 
found  by  actual  experiment : 

Discharges  of  a  6-inch  pipe  at  several  inclinations. 


Inclination. 

Discharges  in  100 
feet  per  minute. 

Inclination. 

Discharges  in  100 
feet  per  minute. 

lin    60 

75 

1  in    320 

49 

lin    80 

68 

1  in    400 

48-5 

1  in  100 

63 

1  in    480 

48 

1  in  120 

59 

1  in    640 

47-5 

1  in  160 

54 

lin    800 

47-2 

1  in  200 

52 

1  in  1200 

46-7 

1  in  240 

50 

Level 

46 

The  conclusion  arrived  at  is,  that  the  requisite  sizes  of  drains 
and  sewers  can  be  determined  (near  enough  for  practical  purposes, 
as  an  important  circumstance  has  to  be  considered  in  providing  for 
the  deposition  of  solid  matter,  which  disadvantageously  alters  the 
form  of  the  aqueduct,  and  contracts  the  water-way)  by  taking  the 
result  of  the  6-inch  pipe,  under  the  circumstances  before  mentioned 
as  a  datum,  and  assuming  that  the  squares  of  the  discharges  are 
as  the  fifth  powers  of  the  diameters. 

That  at  greater  declivities  than  1  in  70,  the  discharges  are  as 
the  square  roots  of  the  inclinations. 


WATER   WHEELS.  327 

That  at  less  declivities  than  1  in  70,  the  usual  law  will  not 
obtain ;  but  near  approximations  to  the  truth  may  be  obtained  by 
observing  the  relative  discharges  of  a  pipe  laid  at  various  small 
inclinations. 

That  increasing  the  number  of  junctions,  at  intervals,  accele- 
rates the  velocity  of  the  main  stream  in  a  ratio  which  increases  as 
the  square  root  of  the  inclination,  and  which  is  greater  than  the 
ratio  of  resistance  due  to  a  proportionable  increase  in  the  length 
of  the  aqueduct.  The  velocity  at  which  the  lateral  streams  enter 
the  main  line,  is  a  most  important  circumstance  governing  the  flow 
of  water.  In  practice,  these  velocities  are  constantly  variable, 
considered  individually,  and  always  different  considered  collectively, 
so  that  their  united  effect  it  is  difficult  to  estimate.  Again,  the 
same  sewer  at  different  periods  may  be  quite  filled,  but  discharges 
in  a  given  time  very  different  quantities  of  water.  It  should  be 
mentioned  that  in  the  case  of  the  6-inch  pipe,  which  discharged 
75  cubic  feet  per  minute,  the  lateral  streams  had  a  velocity  of 
a  few  feet  per  second,  and  the  junctions  were  placed  at  an  angle 
of  about  35°  with  the  main  line.  It  is  needless  to  say  that  all 
junctions  should  be  made  as  nearly  parallel  with  the  main  line  as 
possible,  otherwise  the  forces  of  the  lateral  currents  may  impede 
rather  than  maintain  or  accelerate  the  main  streams. 


WATER  WHEELS. 

THE   UNDERSHOT   WHEEL. 

THE  ratio  between  the  power  and  effect  of  an  undershot  wheel 
is  as  10  to  3-18  ;  consequently  31-43  Ibs.  of  water  must  be  expended 
per  second  to  produce  a  mechanical  effect  equal  to  that  of  the  esti- 
mated labour  of  an  active  man. 

The  velocity  of  the  periphery  of  the  undershot  wheel  should  be 
equal  to  half  the  velocity  of  the  stream  ;  the  float-boards  should  be 
so  constructed  as  to  rise  perpendicularly  from  the  water ;  not  more 
than  one-half  should  ever  be  below  the  surface ;  and  from  3  to  5 
should  be  immersed  at  once,  according  to  the  magnitude  of  the 
wheel. 

The  following  maxims  have  been  deduced  from  experiments : — 

1.  The  virtual  or  effective  head  of  water  being  the  same,  the 
effect  will  be  nearly  as  the  quantity  expended ;  that  is,  if  a  mill, 
driven  by  a  fall  of  water,  whose  virtual  head  is  10  feet,  and  which 
discharges  30  cubic  feet  of  water  in  a  second,  grind  four  bolls  of 
corn  in  an  hour ;  another  mill  having  the  same  virtual  head,  but 
which  discharges  60  cubic  feet  of  water,  will  grind  eight  bolls  of 
corn  in  an  hour. 

2.  The  expense  of  water  being  the  same,  the  effect  will  be  nearly 
as  the  height  of  the  virtual  or  effective  head. 

3.  The  quantity  of  water  expended  being  the  same,  the  effect  is 
nearly  as  the  square  of  its  velocity ;  that  is,  if  a  mill,  driven  by  a 


328  THE   PRACTICAL   MODEL   CALCULATOR. 

certain  quantity  of  water,  moving  with  the  velocity  of  four  feet  per 
second,  grind  three  bolls  of  corn  in  an  hour ;  another  mill,  driven 
by  the  same  quantity  of  water,  moving  with  the  velocity  of  five 
feet  per  second,  will  grind  nearly  4^  bolls  in  the  hour,  because 
3  :  4&  : :  42  :  52  nearly. 

4.  The  aperture  being  the  same,  the  effect  will  be  nearly  as  the 
cube  of  the  velocity  of  the  water ;  that  is,  if  a  mill  driven  by  water, 
moving  through  a  certain  aperture,  with  the  velocity  of  four  feet 
per  second,  grind  three  bolls  of  corn  in  an  hour ;  another  mill, 
driven  by  water,  moving  through  the  same  aperture  with  the  velo- 
city of  five  feet  per  second,  will  grind  5||  bolls  nearly  in  an  hour ; 
for  as  3  :  5|f  : :  43  :  53  nearly. 

The  height  of  the  virtual  head  of  water  may  be  easily  deter- 
mined from  the  velocity  of  the  water,  for  the  heights  are  as  the 
squares  of  the  velocities,  and,  consequently,  the  velocities  are  as 
the  square  roots  of  the  height. 

To  calculate  the  proportions  of  undershot  wheels. — Find  the  per- 
pendicular height  of  the  fall  of  water  above  the  bottom  of  the  mill- 
course,  and  having  diminished  this  number  by  one-half  the  depth  of 
the  water  where  it  meets  the  wheel,  call  that  the  height  of  the  fall. 

Multiply  the  height  of  the  fall,  so  found,  by  64-348,  and  take  the 
square  root  of  the  product,  which  will  be  the  velocity  of  the  water. 

Take  one-half  of  the  velocity  of  the  water,  and  it  will  be  the 
velocity  to  be  given  to  the  float-boards,  or  the  number  of  feet  they 
must  move  through  in  a  second,  to  produce  a  maximum  effect. 
Divide  the  circumference  of  the  wheel  by  the  velocity  of  its  float- 
boards  per  second,  and  the  quotient  will  be  the  number  of  seconds 
in  which  the  wheel  revolves.  Divide  60  by  the  quotient  thus  found, 
and  the  new  quotient  will  be  the  number  of  revolutions  made  by 
the  wheel  in  a  minute. 

Divide  90,  the  number  of  revolutions  which  a  millstone,  5  feet 
in  diameter,  should  make  in  a  minute,  by  the  number  of  revolutions 
made  by  the  wheel  in  a  minute,  the  quotient  will  be  the  number  of 
turns  the  millstone  ought  to  make  for  one  turn  of  the  wheel. 
Then,  as  the  number  of  revolutions  of  the  wheel  in  a  minute  is  to 
the  number  of  revolutions  of  the  millstone  in  a  minute,  so  must 
the  number  of  staves  in  the  trundle  be  to  the  number  of  teeth  in 
the  wheel,  (the  nearest  in  whole  numbers.)  Multiply  the  number 
of  revolutions  made  by  the  wheel  in  a  minute,  by  the  number  of 
revolutions  made  by  the  millstone  for  one  turn  of  the  wheel,  and 
the  product  will  be  the  number  of  revolutions  made  by  the  millstone 
in  a  minute. 

The  effect  of  the  water  wheel  is  a  maximum,  when  its  circum- 
ference moves  with  one-half,  or,  more  accurately,  with  three- 
sevenths  of  the  velocity  of  the  stream. 

THE   BREAST   WHEEL. 

The  effect  of  a  breast  wheel  is  equal  to  the  effect  of  an  under 
shot  wheel,  whose  head  of  water  is  equal  to  the  difference  of  level 


WATER   WHEELS. 


329 


between  the  surface  of  water  in  the  reservoir,  and  the  part  where 
it  strikes  the  wheel,  added  to  that  of  an  overshot,  whose  height  is 
equal  to  the  difference  of  level  between  the  part  where  it  strikes 
the  wheel  and  the  level  of  the  tail  water. 

When  the  fall  of  water  is  between  4  and  10  feet,  a  breast 
wheel  should  be  erected,  provided  there  be  enough  of  water ;  an 
undershot  should  be  used  when  the  fall  is  below  4  feet,  and  an 
overshot  wheel  when  the  fall  exceeds  10  feet.  Also,  when  the  fall 
exceeds  10  feet,  it  should  be  divided  into  two,  and  two  breast  wheels 
be  erected  upon  it. 

TABLE  for  breast  wheels. 


Ireadth  of  the 
float-boards. 

Depth  of  the 
float-boards. 

idius  of  water 
jeel,  reckoned 
n  the  extremity 
r  float-boards. 

elocity  of  the 
wheel  in  a 
second. 

ie  in  which  the 
sel  performs  one 
revolution. 

•ns  of  the  mill- 
,one  for  one  of 
the  wheels. 

ce  of  the  water 
pon  the  float- 
boards. 

ter  required  in 
second  to  turn 
the  wheel. 

R 

t 

«^|o 

K 

H" 

i" 

£* 

Feet. 

Feet. 

Feet. 

Feet. 

Sec. 

Ibs.  avr. 

Cubic  ft. 

1 

0-17 

198-6 

0-75 

2-18 

1-92 

4-80 

1536 

74-30 

2 

0-34 

35-1 

1-50 

3-09 

2-72 

6-80 

1084 

37-15 

3 

0-51 

12-7 

2-26 

3-78 

3-33 

8-32 

886 

24-77 

4 

0-69 

6-2 

3-01 

4-36 

3-84 

9-60 

762 

18-57 

5 

0-86 

3-57 

3-76 

4-88 

4-28 

10-70 

680 

14-86 

6 

1-03 

2-25 

4-51 

5-35 

4-70 

11-76 

626 

12-38 

7 

1-20 

1-53 

5-26 

5-77 

5-08 

12-70 

581 

10-61 

8 

1-37 

1-10 

6-02 

6-17 

5-43 

13-58 

543 

9-29 

9 

1-54 

0-81 

6-77 

6-55 

5-76 

14-40 

512 

8-26 

10 

1-71 

0-77 

7-52 

6-90 

6-07 

15-18 

486 

7-43 

It  is  evident,  from  the  preceding  table,  that  when  the  height  of 
the  fall  is  less  than  3  feet,  the  depth  of  the  float-boards  is  so  great, 
and  their  breadth  so  small,  that  the  breast  wheel  cannot  well  be 
employed ;  and,  on  the  contrary,  when  the  height  of  the  fall  ap- 
proaches to  10  feet,  the  depth  of  the  float-boards  is  too  small  in 
proportion  to  their  breadth ;  these  two  extremes,  therefore,  must 
be  avoided  in  practice.  The  ninth  column  contains  the  quantity 
of  water  necessary  for  impelling  the  wheel ;  but  the  total  expense 
of  water  should  always  exceed  this  by  the  quantity,  at  least,  which 
escapes  between  the  mill-course  and  the  sides  and  extremities  of 
the  float-boards. 

THE   OVERSHOT   WHEEL. 

The  ratio  between  the  power  and  effect  of  an  overshot  wheel,  is 
as  10  to  6*6,  when  the  water  is  delivered  above  the  apex  of  the 
wheel,  and  is  computed  from  the  whole  height  of  the  fall ;  and  as 
10  to  8  when  computed  from  the  height  of  the  wheel  only;  con- 
sequently, the  quantity  of  water  expended  per  second,  to  produce 
a  mechanical  effect  equal  to  that  of  the  aforesaid  estimated  labour 
_of  an  active  man,  is,  in  the  first  instance,  1&-15  Ibs.,  and  in  the 
second  instance,  12-5  Ibs. 

Hence,  the  effect  of  the  ovejshot  wheel,  under  the  same  circum- 
2c2 


330 


THE  PRACTICAL  MODEL  CALCULATOR. 


stances  of  quantity  and  fall,  is,  at  a  medium,  double  that  of  the 
undershot. 

The  velocity  of  the  periphery  of  an  overshot  wheel  should  be 
from  6J  to  8J  feet  per  second. 

The  higher  the  wheel  is,  in  proportion  to  the  whole  descent,  the 
greater  will  be  the  effect. 

And  from  the  equality  of  the  ratio  between  the  power  and  effect, 
subsisting  where  the  constructions  are  similar,  we  must  infer  that 
the  effects,  as  well  as  the  powers,  are  as  the  quantities  of  water  and 
perpendicular  heights  multiplied  together  respectively. 

Working  machinery  by  hydraulic  pressure. — The  vertical  pressure 
of  water,  acting  on  a  piston,  for  raising  weights  and  driving  machi- 
nery, is  coming  into  use  in  many  places  where  it  can  be  advantage- 
ously applied.  At  Liverpool,  Newcastle,  Glasgow,  and  other  places, 
it  is  applied  to  the  working  of  cranes,  drawing  coal-wagons,  and  other 
purposes  requiring  continuous  power.  The  presence  of  a  natural  fall, 
like  that  of  Golway,  Ireland,  which  can  be  conducted  to  the  engine 
through  pipes,  is,  of  course,  the  most  economical  situation  for  the 
application  of  such  power  ;  in  other  situations,  artificial  power  must 
be  used  to  raise  the  water,  which,  even  under  this  disadvantage,  may, 
from  its  readiness  and  simplicity  of  action,  be  often  serviceably  em- 
ployed. Wherever  the  contiguity  of  a  steam  engine  would  be  dan- 
gerous, or  otherwise  objectionable,  a  water  engine  would  afford  the 
means  of  receiving  and  applying  the  power  from  any  required  dis- 
tance, precautions  being  taken  against  the  action  of  frost  on  the  fluid. 

Required  the  horse  power  of  a  centre  discharging  Turbine  water 
wheel,  the  head  of  water  being  25  feet,  and  the  area  of  the  open- 
ing 400  inches. 

The  following  table  shows  the  working  horse  power  of  both  the 
inward  and  outward  discharging  Turbine  water  wheels ;  they  are 
calculated  to  the  square  inch  of  opening. 


Centre  Discharging 
Turbine. 

Outward  Discharg- 
ing Turbine. 

Centre  Discharging 
Turbine. 

Outward  Discharg- 
ing Turbine. 

Head. 

Horse  Power. 

Horse  Power. 

Head. 

Horse  Power. 

Horse  Power. 

3 

•00821 

•012611 

22 

•19523 

•339972 

4 

•01483 

•025145 

23 

•20787 

•364182 

5 

•02137 

•038124 

24 

•22315 

•384615 

6 

•02685 

•045618 

25 

•23667 

•412013 

7 

•03414 

•058314 

26 

•25125 

•437519 

8 

•04198 

•074413 

27 

•26482 

•455698 

9 

•05206 

•089025 

28 

•28135 

•484427 

10 

•05883 

•106215 

29 

•29563 

•510833 

11 

•06921 

•118127 

30 

•30817 

•537721 

12 

•07851 

•135610 

31 

•32316 

•561425 

13 

•08882 

•150638 

32 

•33617 

•587148 

14 

•10054 

•173158 

33 

•34823 

•611013 

15 

•11002 

•192234 

34 

•36154 

•638174 

16 

•12093 

•211592 

35 

•37123 

•665164 

17 

•13196 

•231161 

36 

•89874 

•692156 

18 

•14275 

•257145 

37 

•40118 

•726148 

19 

•15613 

•273325 

38 

•41762 

•764115 

20 

•16927 

•296618 

39 

•42156 

•804479 

21 

•18109 

•317167 

40* 

•43718 

•849814 

WATER   WHEELS. 


331 


Opposite  25  in  the  column  marked  "  Head,"  the  working  horse 
power  to  the  square  inch  is  found  to  be  -25667,  which,  multiplied 
by  400,  gives  94-668,  the  horse  power  required. 

What  is  the  working  horse  power  of  an  outward  discharging 
Turbine,  under  the  effective  head  of  20  feet ;  the  area  of  all  the 
openings  being  325  square  inches.  In  the  table,  opposite  20,  we 
find  -296618,  then  -296618  X  325=  96-4,  the  required  horse  power. 

What  is  the  number  of  revolutions  a  minute  of  an  outward 
discharging  Turbine  wheel,  the  head  being  19  feet  and  the  dia- 
meter of  the  wheel  60  inches  ? 

In  the  table  for  the  outward  discharging  wheel,  opposite  19,  and 
under  60  inches,  we  find  97,  the  number  of  revolutions  required. 

What  is  the  number  of  revolutions  a  minute  of  an  inward  dis- 
charging Turbine,  under  a  head  of  21  feet,  the,  diameter  being 
72  inches  ? 

In  the  table  for  the  inward  discharging  wheel,  opposite  21  feet, 
and  under  72  inches,  we  find  95,  the  number  of  revolutions  a 
minute. 

These  Turbine  tables  were  calculated  by  the  author's  brother, 
the  late  John  O'Byrne,  C.  E.,  who  died  in  New  York,  on  the  6th 
of  April,  1851. 

Outward  discharging  Turbine. 


a 
P 

DIAMETER  IN  INCHES. 

» 

24 

30 

36 

42 

48 

54 

60 

66 

72 

78 

84 

90 

96 

3 

100 

80 

70 

60 

52 

42 

37 

35 

32 

30 

28 

27 

21 

4 

111 

89 

73 

63 

57 

49 

44 

41 

37 

34 

32 

30 

28 

5 

123 

100 

82 

71 

62 

55 

51 

45 

42 

38 

37 

33 

31 

6 

135 

109 

91 

78 

68 

62 

55 

50 

45 

42 

38 

37 

36 

7 

146 

118 

96 

84 

73 

65 

59 

53 

49 

47 

42 

40 

38 

8 

156 

125 

105 

90 

79 

71 

63 

57 

52 

49 

43 

42 

39 

9 

166 

133 

111 

95 

83 

75 

67 

61 

57 

50 

49 

45 

41 

10 

175 

140 

117 

100 

87 

79 

70 

64 

59 

55 

51 

47 

46 

11 

183 

147 

122 

105 

92 

81 

74 

,  67 

62 

57 

54 

49 

48 

12 

191 

156 

127 

110 

96 

85 

79 

70 

64 

59 

55 

53 

51 

13 

200 

159 

133 

115 

100 

89 

81 

73 

67 

62 

57 

55 

53 

14 

206 

166 

138 

118 

104 

92 

83 

75 

69 

64 

59 

57 

55 

16 

213 

171 

142 

122 

107 

95 

86 

78 

72 

66 

61 

58 

56 

16 

222 

177 

148 

126 

111 

98 

89 

82 

74 

69 

64 

59 

57 

17 

227 

182 

152 

131 

115 

101 

91 

83 

77 

71 

66 

62 

59 

18 

234 

187 

156 

134 

117 

105 

94 

85 

78 

73 

67 

63 

61 

19 

238 

193 

161 

138 

120 

107 

97 

88 

81 

74 

69 

64 

63 

20 

247 

197 

164 

141 

124 

110 

99 

90 

84 

76 

71 

66 

64 

21 

252 

202 

168 

145 

126 

114 

101 

92 

85 

78 

73 

68 

65 

22 

259 

208 

172 

149 

129 

115 

105 

94 

87 

80 

74 

69 

67 

23 

263 

212 

176 

151 

133 

119 

106 

96 

89 

84 

77 

72 

70 

24 

270 

216 

180 

155 

135 

120 

109 

98 

92 

85 

78 

•74 

72 

25 

277 

222 

184 

158 

138 

123 

111 

101 

93 

86 

80 

76 

74 

26 

282 

226 

189 

161 

141 

125 

113 

103 

95 

87 

81 

78 

76 

27 

286 

229 

191 

165 

143 

129 

116 

105 

97 

88 

83 

79 

77 

28 

291 

233 

195 

167 

146 

130 

118 

107 

99 

91 

85 

80 

78 

29 

297 

237 

199 

170 

149 

132 

119 

109 

100 

92 

86 

81 

80 

30 

303 

241 

202 

174 

152 

135 

122 

111 

102 

94 

88 

82 

81 

332 


THE  PRACTICAL  MODEL  CALCULATOR. 


Inward  discharging  Turbine. 


Head  in 

feet. 

DIAMETER  IN  MICHES. 

24 

30 

36 

42 

48 

54 

60 

66 

72 

78 

e-± 

90 

96 

3 

111 

86 

74 

62 

54 

48 

47 

40 

36 

32 

31 

30 

27 

4 

125 

96 

83 

70 

62 

55 

51 

45 

41 

37 

36 

34 

31 

6 

141 

112 

94 

78 

69 

61 

55 

50 

46 

43 

40 

37 

36 

6 

152 

122 

101 

86 

76 

67 

62 

55 

51 

47 

43 

42 

38 

7 

166 

131 

108 

93 

82 

72 

65 

60 

54 

51 

47 

44 

42 

8 

175 

139 

116 

99 

87 

76 

71 

63 

57 

54 

49 

47 

45 

9 

186 

149 

123 

06 

93 

81 

74 

68 

63 

57 

53 

51 

47 

10 

195 

156 

129 

111 

99 

86 

78 

71 

66 

61 

56 

52 

49 

11 

208 

167 

136 

117 

102 

91 

82 

74 

68 

63 

58 

56 

52 

12 

217 

169 

142 

122 

107 

97 

85 

78 

71 

66 

61 

57 

54 

13 

221 

178 

148 

127 

112 

99 

89 

82 

74 

69 

64 

61 

56 

14 

231 

184 

153 

133 

116 

104 

92 

85 

76 

71 

66 

62 

58 

15 

238 

191 

159 

136 

119 

107 

95 

87 

80 

73 

68 

64 

61 

16 

245 

198 

165 

144 

123 

111 

99 

90 

83 

76 

71 

66 

63 

17 

252 

203 

168 

148 

127 

114 

102 

92 

85 

78 

73 

68 

64 

18 

269 

209 

173 

150 

132 

116 

104 

95 

87 

82 

75 

69 

66 

19 

267 

215 

176 

153 

134 

120 

108 

98 

89 

83 

77 

72 

67 

20 

276 

222 

183 

157 

138 

122 

111 

101 

93 

85 

79 

74 

69 

21 

288 

226 

186 

162 

141 

125 

113 

103 

95 

86 

80 

75 

71 

22 

290 

230 

192 

164 

145 

129 

116 

107 

96 

89 

83 

77 

73 

23 

299 

235 

196 

167 

146 

133 

118 

109 

97 

91 

84 

79 

74 

24 

303 

240 

201 

171 

151 

135 

122 

111 

101 

93 

86 

80 

75 

25 

310 

247 

206 

176 

155 

138 

123 

112 

104 

96 

88 

82 

76 

26 

314 

248 

210 

180 

157 

139 

126 

115 

106 

97 

90 

84 

79 

27 

319 

254 

213 

183 

162 

142 

128 

117 

108 

99 

92 

85 

80 

28 

327 

261 

218 

186 

164 

146 

129 

119 

109 

102 

93 

87 

82 

29 

333 

265 

221 

189 

166 

148 

133 

121 

111 

103 

95 

89 

83 

30 

336 

271 

224 

193 

168 

151 

136 

124 

114 

105 

97 

90 

85 

WINDMILLS. 

1.  THE  velocity  of  windmill  sails,  whether  unloaded  or  loaded, 
so  as  to  produce  a  maximum  effect,  is  nearly  as  the  velocity  of  tjie 
wind,  their  shape  and  position  being  the  same. 

2.  The  load  at  the  maximum  is  nearly,  but  somewhat  less  than, 
as  the  square  of  the  velocity  of  the  wind,  the  shape  and  position 
of  the  sails  being  the  same. 

3.  The  effects  of  the  same  sails,  at  a  maximum,  are  nearly,  but 
somewhat  less  than,  as  the  cubes  of  the  velocity  of  the  wind. 

4.  The  load  of  the  same  sails,  at  the  maximum,  is  nearly  as  the 
squares,  and  their  effect  as  the  cubes  of  their  number  of  turns  in  a 
given  time. 

5.  When  sails  are  loaded  so  as  to  produce  a  maximum  at  a  given 
velocity,  and  the  velocity  of  the  wind  increases,  the  load  continu- 
ing the  same, — 1st,  the  increase  of  effect,  when  the  increase  of  the 
velocity  of  the  wind  is  small,  will  be  nearly  as  the  squares  of  those 
velocities ;  2dly,  when  the  velocity  of  the  wind  is  double,  the  ef- 
fects will  be  nearly  as  10  to  27  J ;  but,  3dly,  when  the  velocities 
compared  are  more  than  double  of  that  when  the  given  load  pro- 
duces a  maximum,  the  effects  increase  nearly  in  the  simple  ratio 
of  the  velocity  of  the  wind. 


WINDMILLS. 


333 


6.  In  sails  where  the  figure  and  position  are  similar,  and  the  ve- 
locity of  the  wind  the  same,  the  number  of  turns,  in  a  given  time, 
will  be  reciprocally  as  the  radius  or  length  of  the  sail. 

7.  The  load,  at  a  maximum,  which  sails  of  a  similar  figure  and 
position  will  overcome,  at  a  given  distance  from  the  centre  of  mo- 
tion, will  be  as  the  cube  of  the  radius. 

8.  The  effects  of  sails  of  similar  figure  and  position  are  as  the 
square  of  the  radius. 

9.  The  velocity  of  the  extremities  of  Dutch  sails,  as  well  as  of  the 
enlarged  sails,  in  all  their  usual  positions  when  unloaded,  or  even 
loaded  to  a  maximum,  is  considerably  greater  than  that  of  the  wind. 

The  results  in  Table  1  are  for  Dutch  sails,  in  their  common  posi- 
tion, when  the  radius  was  30  feet.  Table  2  contains  the  most 
efficient  angles. 


1. 


2. 


Number  of 
revolutions  of 
wind-shaft  in 
a  minute. 

Velocity  of 
the  wind  in 
an  hour. 

Ratio  between 
velocity  of 
wind  and  re- 
volutions of 
wind-shaft. 

Parts  of  the 
radius,  which 
is  divided  into 
six  parts. 

Angle  with 
the  axis. 

Angle  of  weather. 

3 

2  miles 

0-666 

1 

2 

72° 
71 

18° 
19 

5 

4  miles 

0-800 

3 
4 

72 
74 

18  middle 
16 

6 

5  miles 

0-833 

5 
6 

77} 
83 

i» 

Supposing  the  radius  of  the  sail  to  be  30  feet,  then  the  sail  will 
commence  at  ^,  or  5  feet  from  the  axis,  where  the  angle  of  inclina- 
tion will  be  72  degrees ;  at  f,  or  10  feet  from  the  axis,  the  angle 
will  be  71  degrees,  and  so  on. 

Results  of  Experiments  on  the  effect  of  Windmill  Sails  in  grind- 
ing corn. — By  M.  COULOMB. 

A  windmill,  with  four  sails,  measuring  72  feet  from  the  ex- 
tremity of  one  sail  to  that  of  the  opposite  one,  and  6  feet  7  inches 
wide,  or  a  little  more,  was  found  capable  of  raising  1100  Ibs.  avoir- 
dupois 238  feet  in  a  minute,  and  of  working,  on  an  average,  eight 
hours  in  a  day.  This  is  equivalent  to  the  work  of  34  men,  30  square 
feet  of  canvas  performing  about  the  daily  work  of  a  man. 

When  a  vertical  windmill  is  employed  to  grind  corn,  the  mill- 
stone makes  5  revolutions  in  the  same  time  that  the  sails  and  the 
arbor  make  1. 

The  mill  does  not  begin  to  turn  till  the  velocity  of  the  wind  is 
about  13  feet  per  second. 

When  the  velocity  of  the  wind  is  19  feet  per  second,  the  sails 
make  from  11  to  12  turns  in  a  minute,  and  the  mill  will  grind  from 
880  to  990  Ibs.  avoirdupois  in  an  hour,  or  about  22,000  Ibs.  in  24 
hours. 


334 


THE  APPLICATION  OF  LOGARITHMS. 


THE  practice  of  performing  calculations  by  Logarithms  is  an  ex- 
ercise so  useful  to  computers,  that  it  requires  a  more  particular  ex- 
planation than  could  have  been  properly  given  in  that  part  of  the 
work  allotted  to  Arithmetic. 

A  few  of  the  various  applications  of  logarithms,  best  suited  to 
the  calculations  of  the  engineer  and  mechanic,  have  therefore  been 
collected,  and  are,  with  other  matter,  given,  in  hopes  that  they  will 
come  into  general  use,  as  the  certainty  and  accuracy  of  their  re- 
sults can  be  more  safely  relied  upon  and  more  easily  obtained 
than  with  common  arithmetic. 

By  a  slight  examination,  the  student  will  perceive,  in  some  de- 
gree, the  nature  and  effect  of  these  calculations ;  and,  by  frequent 
exercise,  will  obtain  a  dexterity  of  operation  in  every  case  admitting 
of  their  use.  He  will  also  more  readily  penetrate  the  plans  of  the 
different  devices  employed  in  instrumental  calculations,  which  are 
rendered  obscure  and  perplexing  to  most  practical  men  by  their  ig- 
norance of  the  proper  application  of  logarithms. 

Logarithms  are  artificial  numbers  which  stand  for  natural  num- 
bers, and  are  so  contrived,  that  if  the  logarithm  of  one  number  be 
added  to  the  logarithm  of  another,  the  sum  will  be  the  logarithm 
of  the  product  of  these  numbers  ;  and  if  the  logarithm  of  one  num- 
ber be  taken  from  the  logarithm  of  another,  the  remainder  is  the 
logarithm  of  the  latter  divided  by  the  former  ;  and  also,  if  the  loga- 
rithm of  a  number  be  multiplied  by  2,  3,  4,  or  5,  &c.,  we  shall  have 
the  logarithm  of  the  square,  cube,  &c.,  of  that  number ;  and,  on  the 
other  hand,  if  divided  by  2,  3,  4,  or  5,  &c.,  we  have  the  logarithm 
of  the  square  root,  cube  root,  fourth  root,  &c.,  of  the  proposed  num- 
ber ;  so  that  with  the  aid  of  logarithms,  multiplication  and  division 
are  performed  by  addition  and  subtraction ;  and  the  raising  of 
powers  and  extracting  of  roots  are  effected  by  multiplying  or  di- 
viding by  the  indices  of  the  powers  and  roots. 

In  the  table  at  the  end  of  this  work,  are  given  the  logarithms  of 
the  natural  numbers,  from  1-  to  1000000  by  the  help  of  differences ; 
in  large  tables,  only  the  decimal  part  of  the  logarithm  is  given,  as 
the  index  is  readily  determined  ;  for  the  index  of  the  logarithm  of 
any  number  greater  than  unity,  is  equal  to  one  less  than  the  num- 
ber of  figures  on  the  left  hand  of  the  decimal  point ;  thus, 

The  index  of  12345-  is  4-, 

1234-5  -  3-, 

12345  -  2-, 

1-  12-345  -  1-, 

1-2345  -  0- 


THE   APPLICATION   OF   LOGARITHMS.  335 

The  index  of  any  decimal  fraction  is  a  negative  number  equal  to 
one  and  the  number  of  zeros  immediately  following  the  decimal 
point ;  thus, 

The  index  of  -00012345  is  -4-  or  4- 

-0012345    is  -3-  or  3; 

-012345      is  -2-  or  2; 

-12345        is  -1-  or  1- 

Because  the  decimal  part  of  the  logarithm  is  always  positive,  it 
is  better  to  place  the  negative  sign  of  the  index  above,  instead  of 
before  it;  thus,  3~-  instead  of  —3.  For  the  log.  of  -00012345  is 
better  expressed  by  ¥-0914911,  than  by  —4-0914911,  because  only 
the  index  is  negative — i.  e.9  4  is  negative  and  -0914911  is  positive, 
and  may  stand  thus,  —4-  -f  -0914911. 

Sometimes,  instead  of  employing  negative  indices,  their  comple- 
ments to  10  are  used  : 

for  ¥-0914911  is  substituted  6-0914911 

—  3^-0914911 7-0914911 

—  1-0914911 8-0914911 

&c.  &c. 

When  this  is  done,  it  is  necessary  to  allow,  at  some  subsequent 
stage,  for  the  tens  by  which  the  indices  have  thus  been  increased. 

It  is  so  easy  to  take  logarithms  and  their  corresponding  numbers 
out  of  tables  of  logarithms,  that  we  need  not  dwell  on  the  method 
of  doing  so,  but  proceed  to  their  application. 

MULTIPLICATION   BY  LOGARITHMS. 

Take  the  logarithms  of  the  factors  from  the  table,  and  add  them 
together ;  then  the  natural  number  answering  to  the  sum  is  the 
product  required :  observing,  in  the  addition,  that  what  is  to  be 
carried  from  the  decimal  parts  of  the  logarithms  is  always  positive, 
and  must  therefore  be  added  to  the  positive  indices  ;  the  difference  be- 
tween this  sum  and  the  sum  of  the  negative  indices  is  the  index  of  the 
logarithm  of  the  product,  to  which  prefix  the  sign  of  the  greater. 

This  method  will  be  found  more  convenient  to  those  who  have 
only  a  slight  knowledge  of  logarithms,  than  that  of  using  the  arith- 
metical complements  of  the  negative  indices. 

1.  Multiply  37-153  by  4-086,  by  logarithms. 

Nbs.  Logs. 

37-153 1-5699939 

4-086 .0-6112984 

Prod.  151-8071 2-1812923 

2.  Multiply  112-246  by  13-958,  by  logarithms. 

Nos.  Logs. 

112-246 2-0501709 

13-958 '.1-1448232 

Prod.  1566-729 ..  ,..3-1949941 


336  THE   PBACTICAL    MODEL    CALCULATOR. 

3.  Multiply  46-7512  by  -3275,  by  logarithms. 

Nbs.  Logs. 

46-7512 1-6697928 

•3275 1-5152113 

Prod.  15-31102 1-1850041 

Here  the  +1  that  is  to  be  carried  from  the  decimals,  cancels 
the  — 1,  and  consequently  there  remains  1  in  the  upper  line  to  be 
set  down. 

4.  Multiply  -37816  by  -04782,  by  logarithms. 

Nos.  Logs. 

•37816 T-5776756 

•04782 2-6796096 

Prod.  0-0180836 ."2-2572852 

Here  the  +1  that  is  to  be  carried  from  the  decimals,  destroys 
the  —1  in  the  upper  line,  as  before,  and  there  remains  the  —2 
to  be  set  down. 

5.  Multiply  3-768,  2-053,  and  -007693,  together. 

Nos.  Logs. 

3-768 0-5761109 

2-053 0-3123889 

•007693 3~-8860957 

Prod.  -0595108 .2-7745955 

Here  the  +1  that  is  to  be  carried  from  the  decimals,  when  ad- 
ded to  —3,  makes  —2  to  be  set  down. 

6.  Multiply  3-586,  2-1046,  -8372,  and  -0294,  together. 

Nos.  Logs. 

3-586 0-5546103 

2-1046 0-3231696 

•8372 .1-9228292 

•0294 ¥-4683473 

Prod.  -1857618 1-2689564 

Here  the  4-2  that  is  to  be  carried,  cancels  the  — 2,  and  there 
remains  the  —1  to  be  set  down. 

DIVISION  BY  LOGARITHMS. 

From  the  logarithm  of  the  dividend,  subtract  the  logarithm  of 
the  divisor ;  the  natural  number  answering  to  the  remainder  will  be 
the  quotient  required. 

Observing,  that  if  the  index  of  the  logarithm  to  be  subtracted  is 
positive,  it  is  to  be  counted  as  negative,  and  jf  negative,  to  be  con- 
sidered as  positive  ;  and  if  one  has  to  be  carried  from  the  decimals, 
it  is  always  negative :  so  that  the  index  of  the  logarithm  of  the 
quotient  is  equal  to  the  sum  of  the  index  of  the  dividend,  the  index 


THE   APPLICATION    OF   LOGARITHMS.  337 

of  the  divisor  with  its  sign  changed,  and   —1  when  1  is  to  be 
carried  from  the  decimal  part  of  the  logarithms. 

1.  Divide  4768-2  by  36-954,  by  logarithms. 

Nos.  Logs. 

4768-2 3-6783545 

36-954 1-5676615 

Quot.  129-032 2-1106930 

2.  Divide  21-754  by  2-4678,  by  logarithms. 

Nos.  Logs. 

21-754 1-3375391 

2-4678.... 0-3923100 

Quot.  8-81514 0-9452291 

3.  Divide  4-6257  by  -17608,  by  logarithms. 

Nos.  Logs. 

4-6257 0-6651775 

-17608 T-2457100 

Quot.  26-27045 1-4194675 

Here  the  —1  in  the  lower  index,  is  changed  into  +1,  which  is 
then  taken  for  the  index  of  the  result. 

4.  Divide  -27684  by  5-1576,  by  logarithms. 

Nos.  Logs. 

•27684 .T-4422288 

5-1576 : 0-7124477 

Quot.  -0536761 "2-7297811 

Here  the  1  that  is  to  be  carried  from  the  decimals,  is  taken  as 
— 1,  and  then  added  to  —1  in  the  upper  index,  which  gives  —2 
for  the  index  of  the  result. 

5.  Divide  6-9875  by  -075789,  by  logarithms. 

Nos.  Logs. 

6-9875 ._0-8443218 

•075789 ¥-8796062 

Quot.  92-1967 1-9647156 

Here  the  1  that  is  to  be  carried  from  the  decimals,  is  added  to 
—2,  which  makes  — J.,  and  this  put  down,  with  its  sign  changed, 
is  +1. 

6.  Divide  -19876  by  -0012345,  by  logarithms. 

Nos.  Logs. 

•19876 A 1:2983290 

•0012345 3-0914911 

Quot.  161-0043 2-2068379 

Here  —3  in  the  lower  index,  is  changed  into  +3,  and  this  ad- 
ded to  1,  the  other  index,  gives  +3  —  1,  or  2. 
2  D  22 


338  THE   PRACTICAL   MODEL   CALCULATOR. 

PROPORTION;    OR,   THE   RULE   OF    THREE,   BY  LOGARITHMS. 

From  the  sum  of  the  logarithms  of  the  numbers  to  be  multiplied 
together,  take  the  sum  of  the  logarithms  of  the  divisors :  the  re- 
mainder is  the  logarithm  of  the  term  sought. 

Or  the  same  may  be  performed  more  conveniently,  for  any 
single  proportion,  thus : — Find  the  complement  of  the  logarithm 
of  the  first  term,  or  what  it  wants  of  10,  by  beginning  at  the  left 
hand  and  taking  each  of  the  figures  from  9,  except  thfe  last  figure 
on  the  right,  which  must  be  taken  from  10 ;  then  add  this  result 
and  the  logarithms  of  the  other  two  figures  together:  the  sum, 
abating  10  in  the  index,  will  be  the  logarithm  of  the  fourth  term. 

1.  Find  a  fourth  proportional  to  37*125,  14-768,  and  135-279, 
by  logarithms. 

Log.  of  37-125 1-5696665 

Complement 8-4303335 

Log.  of  14-768 1-1693217 

Log.  of  135-279 2-1312304       I 

Ans.  53-8128 1-7308856 

2.  Find  a  fourth  proportional  to  -05764,  -7186,  and  -34721,  by 
logarithms. 

Log.  of  -05764 2-7607240 

Complement 11-2392760 

Log.  of  -7186..'. 1-8564872 

Log.  of -34721 1-5405922 

Ans.  4-32868 s 0-6363554 

3.  Find  a  third  proportion^  to  12-796  and  3-24718,  by  logarithms, 

Log.  of  12-796. 1-1070742 

Complement 8-8929258 

Log.  of  3-24718 0-5115064 

Log.  of  3-24718 0-5115064 

Ans.  -8240216 T-9159386 

INVOLUTION;   OR,    THE   RAISING  OF   POWERS,   BY  LOGARITHMS. 

Multiply  the  logarithm  of  the  given  number  by  the  index  of 
the  proposed  power ;  then  the  natural  number  answering  to  the 
result  will  be  the  power  required.  Observing,  if  the  index  be  nega- 
tive, the  index  of  the  product  will  be  negative  ;  but  as  what  is  to 
be  carried  from  the  decimal  part  will  be  affirmative,  therefore  the 
difference  is  the  index  of  the  result. 

1.  Find  the  square  of  2-7568,  by  logarithms. 

Log.  of  2-7568 0-4404053 

2 

Square  7-599947 0-8808106 


THE  APPLICATION   OF  LOGARITHMS.  339 

2,  Find  the  cube  of  7-0851,  by  logarithms. 

Log.  of  7-0851 0-8503460 

3 


Cube  355-6625 2-5510380 

Therefore  355-6625  is  the  answer. 

3.  Find  the  fifth  power  of  -87451,  by  logarithms. 

Log.  of  -87451 r-9417648 

o 

Fifth  power  -5114695 1-7088240 

Where  5  times  the  negative  index  1,  being  —5,   and  +4  to 
carry,  the  index  of  the  power  is  1. 

4.  Find  the  365th  power  of  1-0045,  by  logarithms. 

Log.  of  1-0045 0-0019499 

<  365 


97495 
116994 

58497 

Power  5-148888 Log.  0-7117135 

EVOLUTION;   OR,   THE  EXTRACTION  OF  ROOTS,   BY  LOGARITHMS. 

Divide  the  logarithm  of  the  given  number  by  2  for  the  square 
root,  3  for  the  cube  root,  &c.,  and  the  natural  number  answering 
to  the  result  will  be  the  root  required. 

But  if  it  be  a  compound  root,  or  one  that  consists  both  of  a  root 
and  a  power,  multiply  the  logarithm  of  the  given  number  by  the 
numerator  of  the  index,  and  divide  the  product  by  the  denomina- 
tor, for  the  logarithm  of  the  root  sought. 

Observing,  in  either  case,  when  the  index  of  the  logarithm  is 
negative,  and  cannot  be  divided  without  a  remainder,  to  increase 
it  by  such  a  number  as  will  render  it  exactly  divisible ;  and  then 
carry  the  units  borrowed,  as  so  many  tens,  to  the  first  figure  of  the 
decimal  part,  and  divide  the  whole  accordingly. 

1.  Find  the  square  root  of  27-465,  by  logarithms. 

Log.  of  27-465 2 )  1-4387796 

Root  5-2407 -7193898 

2.  Find  the  cube  root  of  35-6415,  by  logarithms. 

Log.  of  35-6415 3 )  1-5519560 

Root  3-29093 ! -5173186 

3.  Find  the  fifth  root  of  7-0825,  by  logarithms. 

Log.  of  7-0825 5 )  0-8501866 

Root  1-479235...  ,..-1700373 


340  THE   PRACTICAL   MODEL   CALCULATOR. 

4.  Find  the  365th  root  of  1-045,  by  logarithms. 

Log.  of  1-045 365)0-0191163 

Koot  1-000121 .0-0000524 

5.  Find  the  value  of  (-001234)^,  by  logarithms. 

Log.  of  -001234 3~-0913152 

2 

3)6-1826304 

Ans.  -00115047 ."2-0608768 

Here  the  divisor  3  being  contained  exactly  twice  in  the  negative 
index  —6,  the  index  of  the  quotient,  to  be  put  down,  will  be  —2. 

6.  Find  the  value  of  (-024554)^,  by  logarithms. 

Log.  of -024554 ."2-3901223 

3 

2)^1703669 

Ans.  -00384754 ."3-5851834 

Here,  2  not  being  contained  exactly  in  —5,  1  is  added  to  it, 
which  gives  —3  for  the  quotient ;  and  the  1  that  is  borrowed  being 
carried  to  the  next  figure  makes  11,  which,  divided  by  2,  gives 
•5851834  for  the  decimal  part  of  the  logarithm. 


AND   THE   NUMBER  CORRESPONDING  TO  ANY   GIVEN  LOGARITHM.      DIS- 
COVERED   BY   OLIVER    BYRNE,    THE   AUTHOR    OF    THE   PRESENT   WORK. 

The  succeeding  numbers  possess  a  particular  property,  which  is 
worth  being  remembered. 

log.  1-371288574238542  =  0-1371288574238542 
log.  10-00000000000000  =  1-000000000000000 
log.  237-5812087593221  =  2-375812087593221 
log.  3550-260181586591  =  3-550260181586591 
log.  46692-46832877758  =  4-669246832877758 
log.  576045-6934135527  =  5-760456934135527 
log.  6834720-776754357  =  6-834720776754357 
log.  78974890-31398144  =  7-897489031398144 
log.  895191599-8267852  =  8-951915998267839 
log.  9999999999-999999  =  9-999999999999999 
In  these  numbers,  if  the  decimal  points  be  changed,  it  is  evident 
the  logarithms  corresponding  can  also  be  set  down  without  any  cal- 
culation whatever. 

Thus,  the  log.  of  137-1288574238542  =  2-1371288574238542; 
the  log.  of  35-50260181586591  =  1^50260181586591; 
log.  -002375812087593221  '=  3-375812087593221 ; 
log. -0008951915998267852  =  1-951915998267852 ; 


THE  APPLICATION   OF   LOGARITHMS. 


341 


and  so  on  in  similar  cases,  since  the  change  of  the  decimal 
point  in  a  number  can  only  affect  the  whole  number  of  its  loga- 
rithm. 

These  numbers  whose  logarithms  are  made  up  of  the  same  digits 
will  be  found  extremely  useful  hereafter.  We  shall  next  give  a 
simple  method  of  multiplying  any  number  by  any  power  of  11, 101, 
1001,  10001,  100001,  &c. 

This  multiplication  is  performed  by  the  aid  of  coefficients  of  a 
binomial  raised  to  the  proposed  power. 

(x  +  y)1  —  x  -f  y,  the  coefficients  are  1,  1. 

(x  +  y)2  =  x2  +  Zxy  +  y\  the  coefficients  are  1,  2,  1. 

(x  +  yf  =  x3  +  3x*y  +  3^2  +  y\  the  coefficients  are  1,  3,  3  1. 

The  coefficients  of  (a  +  y}*  are  1,  4,  6,  4,  1. 

—  —          (x  +  y)5—  1,  5,  10,  10,  5,  1. 

—  —         (x  +  y)«—  1,  6,  15,  20,  15,  6,  1. 

(x  +  y)'—  1,7,21,35,35,21,7,1. 
-         (*  +  ^)8-  1,  8,  28,  56,  70,  56,  28,  8,  1. 

—  —         (x  +  y)»  —  1,  9,  36,  84, 126, 126,  84,  36,  9, 1. 

Let  it  be  required  to  multiply  54247  by  (101)6. 

The  number  must  be  divided  into  periods  of  two  figures  when  the 
nultiplier  is  101 ;  into  periods  of  three  figures  when  the  multiplier 
.8  1001 ;  into  periods  of  four  figures  when  the  multiplier  is  10001 ; 
ind  so  on. 


a 

c 

b 

a 

54 

24 

70 

00 

00 

1 

3 

25 

48 

20 

00 

a  6 

8 

13 
10 

70 

84 

50 
94 

b  15 
o  20 

8 

14 

d  15 

3 

e  6 

54247)  x  (101)6  =  57  58  42  83  61,  true  to  10  places  of  figures. 

This  operation  is  readily  understood,  since  the  multipliers  for  the 
>th  power  are  1,  6, 15,  20, 15,  6,  1 ;  we  begin  at  a9  a  period  in  ad- 
vance, arid  multiply  by  6 ;  then  we  commence  at  5,  two  periods  in 
idvance,  and  multiply  by  15 ;  at  c,  three  periods  in  advance,  and 
nultiply  by  20 ;  at  d,  four  periods  in  advance  (counting  from  the 
•ight  to  the  left),  and  multiply  by  15 ;  the  period,  e,  should  be 
nultiplied  by  6,  but,  as  it  is  blank,  we  only  set  down  the  3  carried 
•rom  multiplying  d,  or  its  first  figure  by  6. 

As  it  is  extremely  easy  to  operate  with  1,  5,  10,  10,  5,  1,  the 
nultipliers  for  the  5th  power,  it  may  be  more  convenient  first  to 
nultiply  the  given  number  by  (101)5,  and  then  by  (101)1 ;  because, 
;o  multiply  any  number  by  5,  we  have  only  to  affix  a  cipher  (or 
suppose  it  affixed)  and  to  take  the  half  of  the  result. 

The  above  example,  if  worked  in  the  manner  just  described,  will 
stand  as  follows : 


342 


THE  PRACTICAL  MODEL  CALCULATOR. 


d 

54 

2 

G 

24 
71 
5 

b 
70 
23 
42 
5 

a 

00 
50 

47 

42 

o 

— 

00 
00 
00 
47 
71 
1 

1 
5..a 
...10..6 
...10..<? 
5..d 
1 

(54247)  x  (101)5  =  57 

01 
57 

41142 
0141 

19 
42 

5758428361  =  (54247)6  x  (101)6. 

The  truth  of  this  is  readily  shown  by  common  multiplication,  but 
the  process  is  cumbersome.  However,  for  the  sake  of  comparison, 
we  shall  in  this  instance  multiply  54247  by  (101)  raised  to  the  6th 
power. 

101 

101 


10201  =  (101)2. 
101 


10201 
102010 

1030301  =  (101)3. 
101 

1030301 
10303010 

104060301  =  (101)4. 
101 

104060401 
1040604010 

10510100501  =  (101)*. 
101 

10510100501 
105101005010 

1061520150601  =  (101)6. 
54247 


7430641054207 
4246080602404 
2123040301202 
4246080602404  ' 
5307600753005 

5758428360(9652447  the  required  product, 


THE   APPLICATION   OF   LOGARITHMS. 


343 


which  shows  that  the  former  process  gives  the  result  true  to  10 
places  of  figures,  of  which  we  shall  add  another  example. 

Multiply  34567812  by  (1001)8,  so  that  the  result  may  be  true  to 
12  places  of  figures. 

b       a    I 
3456  7812  0000 
2  7654  2496 
96790 
19 


....1 

S..a 

...2S..b 


3459  5475  9305  the  required  product. 

The  remaining  multipliers,  70,  56,  28,  8,  1,  are  not  necessary  in 
obtaining  the  first  12  figures  of  the  product  of  34567812  by  10001 
in  the  8th  power. 

A's  28  and  56  are  large  multipliers,  the  work  may  stand  thus 


b 

a 

3456 

7812 

0000 

1 

2 

7654 

2496 

...a..  8 

6 
2 

9136 
7654 

...5..2(T 
...b..  8> 

>28 

17 

2 

...o.'.  6d 

j>56 

Result,  =  345954759305  the  same  as  before. 

Perhaps  this  product  might  be  obtained  with  greater  ease  by  first 
multiplying  34567812  by  (10001)5,  and  the  product  by  (10001)3 ; 
the  operation  will  stand  thus : 

345678120000 1 

172839060 5 

34568 10 

3 10 

345850093631  =  34567812  x  (10001)5. 

103755298 3 

10376...    .  3 


345954759305  =  twelve   places   of  the   product   of 
34567812  by  (10001)5  x  (10001)3  =  (34567812)  x  (10001)8. 

Although  these  methods  are  extremely  simple,  yet  cases  will  oc- 
cur, when  one  of  them  will  have  the  preference. 

Our  next  object  is  to  determine  the  logarithms  1*1 ;  1*01 ;  1*001-; 
1-0001;  1-00001;  &c. 

It  is  well  known  that 

log.  (1  +  n)  =  M  (n  -  \tf  +  %n3  -  \n*  +  \ns  —  K  +  &c.) 
M  being  the  modulus,  =  -432944819032618276511289,  &c. 

It  is  evident  that  when  n  is  ^,  ^  j^Vo?  ioo<n>>  &c-»  tae  calcula- 
tion becomes  yery  simple. 


344  THE   PRACTICAL   MODEL   CALCULATOR. 

M  =  -4342944819032518 
}  M  =  -2171472409516259 
|  M  =  -1447648273010839 
J  M  =  -1085736204758130 
|  M  =  -0868588963806504 
}  M  =  -0723824136505420 
*M  =  -0720420788433217 
|  M  =  -0542868102379065 
i  M  =  -0482549424336946 
-0434294481903252 


&c.  &c.,  are  constants  employed  to  determine 
the  logarithms  of  11,  101,  1001;  100001,  &c. 

To  compute  the  log.  of  1-001.     In  this  case  n  =  ^fa. 
+  =  -0004342944819033  positive 


'000000^171472410  negative 
•0004340773346623 

'0000000001447648  positive 
•0004340774794271 


negative 


•0004340774793185 


=  -0000000000000001  positive 

•0004340774793186  =  the  log.  of  1-001  ; 
true  to  sixteen  places. 

It  is  almost  unnecessary  to  remark,  that,  instead  of  adding  and 
subtracting  alternately,  as  above,  the  positive  and  negative  terms 
may  be  summed  separately,  which  will  render  the  operation  more 
concise. 


Negative  Terms. 

•0000002171472410 
1086 

•0000002171473496 


Positive  Terms. 
•0004342944819033 
1447648 
1 

+  -0004342945266682 
-     OQQOOQ217473496 

•0004340774793186  =  log.  1-001. 

In  a  similar  manner  the  succeeding  logarithms  may  be  obtained 
to  almost  any  degree  of  accuracy. 


THE   APPLICATION   OT   LOGARITHMS. 


345 


Log.  1-1 
1-01 
1-001 
1-0001 
1-00001 
1-000001 
1-0000001 
1-00000001 
1-000000001 
1-0000000001 
1-00000000001 
1-000000000001 
1-0000000000001 
1-00000000000001 
&c. 


=  -041392685158225  &c.  which  we  call  A 


•004321373782643 
•000434077479319 
•000043427276863 
•000004342923104 
•000000434294265 
•000000043429447 
•000000004342945 
•000000000434295 
•000000000043430 
•000000000004343 
•000000000000434 
•000000000000043 
•000000000000004 
&c. 


B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
M 


&c. 


Without  further  formality  or  paraphernalia,  for  it  is  presumed 
that  such  is  not  necessary,  we  shall  commence  operating,  as  the 
method  can  be  acquired  with  ease,  and  put  in  a  clearer  point  of 
view  by  proper  examples. 

Required  the  logarithm  of  542470,  to  seven  places  of  decimals. 


54 

24 

70 

. 

V3 

25 

48 

20 

13 

71 

10 

85 

8 

57584284  =  6B  =  -02592824 

7275 


Take  57  6  01  56  2  =  3D 
From57604569 


576) 


007 

880  =  5E 


•00013028. 


•00002171 


27 
15 


2  = 


=  -00000087 


=  -00000009 

•02608119  Take 
5-76045693  From 


Hence  we  have  log.  542470  =  5-73437574,  which  is  correct 
to  seven  decimal  places. 

6B  is  written  to  represent  6  times  the  log.  of  1*01. 

The  nearest  number  to  542470,  whose  log.  is  composed  of  the 
same  digits  as  itself,  being  576045*6934,  &c.,  our  object  was  to 
raise  542470-  to  576045-69  by  multiplying  542470-  by  some  power 
or  powers  of  1-1,  1-01,  1-001,  1-0001,  &c. 


346 


THE  PRACTICAL  MODEL  CALCULATOR. 


It  is  here  necessary  to  remark,  that  A  is  not  employed,  because 
the  given  number  multiplied  by  1-1,  would  exceed  576045-69 ;  for 
a  like  reason  C  is  omitted. 

Again,  when  half  the  figures  coincide,  the  process  may  be  per- 
formed (as  above)  by  common  division ;  the  part  which  coincides 
becoming  the  divisor ;  thus,  in  finding  5  E,  576  is  divided  into  3007, 
it  goes  5  times,  the  E  showing  that  there  are  five  figures  in  each 
period  at  this  step.  For  A,  there  is  but  one  figure  in  each  period  ; 
for  B,  there  are  two  figures ;  for  C,  there  are  three  figures  in  each 
period,  and  so  on. 

Let  it  be  required  to  calculate  the  logarithm  of  2785-9,  true  to 
seven  places  of  decimals. 

It  will  be  found  more  convenient,  in  this  instance,  to  bring  the 
given  number  to  3550-26018,  the  log.  of  which  is  3-55026908. 


27 
5 

859 
571 

278 

0 

8 
5 

0 
0 
9 

0 
0 
0 

33 

1 

7093 
6854 
337 
3 

Q  O  O  A 

70 
09 
37 

2 

35 

4289 
708 

08  =  5B 
58 
35 

Take35499801  =  20 
From  35502602 

355)  •  • 

•  -2 
2 

801  =  7E 

485 

3 

2 

16  =  8F 

84 

3 
3 

2  =*  9G 
2 

=  2  A  =  -08278537 


'02160687 


=  -00086815 


•00003040 


8F  =  -00000347 


=  -00000039 


Take     -10529465 
From  3-55026018 


log.  2785-9  =  3-44496553 

At  the  Observatory  at  Paris,  g  —  9-80896  metres,  the  second 
being  the  unit  of  time,  what  is  the  logarithm  of  9-80896  ? 
In  this  example,  we  shall  bring  9-80896  to  9-99999,  &c. 


THE   APPLICATION   OF   LOGARITHMS. 


347 


98 

08 
98 

96(00 

08|96 

00 
00 

990j704 
8i916 


960 
344 


356654 

83 


0  =  1 B  =  -0043213738 
6 


999615705 
2:9989 


32  =  90  =  -0039066973 

72 
00 


9999569804 

399983 

6 


3  D  =  -0001302818 


Take     9999969793  =  4E=  -0000173717 
From  1  0000  QOfrQOO 

.....  30207 

From  which  we  have  .........  3F  =  -0000013029 

2  H  =  -0000000087 
7J  =  -0000000003 


-0083770365 
From  1-0000000000 

Log.  9-80896  =  -9916229635 

As  before  observed,  9  C  might  have  been  obtained  in  the  follow- 
ing manner  : 

0  =  1  B,  as  above. 


5  times 

890 
4 

704 
953 
9 

960 
524 
907 
9 

995 
3 

668 
982 
5 

401 
673 
973 
4 

4times9996570532  =  9C. 

A  French  metre  is  equal  to  3-2808992  English  feet,  required 
the  log.  of  3-2808992. 


eld 

c 

b 

a 

32 

80 

89 

92 

00 

...once 

1  2 

29 

66 

29 

44 

...  7 

times  from  a 

6 

88 

98 

88 

...21 

—        b 

11 

48 

31 

...35 

—         c 

11 

48 

...35 

—       d 

7 

...21 

—         e 

3517568018  =  B  7. 


348  THE    PRACTICAL   MODEL   CALCULATOR. 

The  manner  in  which  B  7  is  obtained  is  worthy  of  remark :  the 
multipliers  being  1,  7,  21,  35,  35,  21,  7,  1,  when  7  times  the  first 
line  (commencing  with  the  period  marked  a)  is  obtained,  21  times 
the  same  line  (commencing  with  the  period  marked  b)  is  determined 
by  multiplying  the  2d  line  by  3.  If  the  2d  line  be  again  multiplied 
by  5,  we  have  the  4th  line  of  the  multiplier  35 ;  but  to  multiply 
by  5,  we  have  only  to  take  the  half  the  product  produced  by  mul- 
tiplying by  7,  advancing  the  result  one  figure  to  the  right.  Hence, 
to  find  the  result  for  35  is  almost  as  easy  as  to  find  the  result 
for  5. 

But  the  object  in  this  case  being  to  bring  the  proposed  number 
to  35502601815,  the  process  must  be  continued. 


1        351 
9  3 


756 


801 


165811 


8  =  B  7,  as  above. 


36  12  663  2 

84       ' 296 

354  935  305  8  =  C  9 

The  2d  (or  9)  line  is  produced  by  beginning  at  a,  but  the  multi- 
plication may  be  performed  by  subtracting  3517568  from  35175680 ; 
the  36  line  is  produced  by  beginning  at  5,  observing  to  carry  from 
the  preceding  figure,  making  the  usual  allowance  when  the  number 
is  followed  by  5,  6,  7,  8,  or  9.  The  36  line  may  be  produced  by 
multiplying  the  9  line  by  4,  beginning  one  period  more  to  the  left. 
To  multiply  by  84  is  not  apparently  so  convenient,  for  84  x  352  = 
29|568  ;  and  as  only  one  figure  of  the  period  568  is  required,  when 
the  proper  allowance  is  made,  the  result  becomes  2916. 

But,  since  84  is  equal  to  36  X  2J-,  we  have  only  to  multiply  the 
36  line  by  2,  and  add  J  of  it ;  with  such  management,  the  work 
will  stand  thus  : — 


351 


756 


3 165  811 


801 


8  =  B  7,  as  before 
2=9  times 


12  663  2  =  36  times 

24  3  =  72  times  \       ft .  . . 
42  =  12  times}  =  84  times 

'—.  9 


354  935  305  8  =  C  9 

This  amounts  to  very  little  more  than  adding  the  above  numbers 
together. 

Many  other  contractions  will  suggest  themselves,  when  the  mul- 
pliers  are  large :  thus,  to  multiply  any  number  57837  by  9,  as 
alluded  to  above,  is  easily  effected,  by  the  following  well-known 
process : — Subtract  the  first  figure  to  the  right  from  10,  the  second 
from  the  first,  the  third  from  the  second,  and  so  on. 

(578370. ..ten  times 
57837. ..once 
520533... nine  times 


THE   APPLICATION   OF   LOGARITHMS. 


349 


Such  simple  observations  are  to  be  found  in  every  book  on  men- 
tal arithmetic,  and  therefore  require  but  little  attention  here. 
The  whole  work  of  the  previous  example  will  stand  thus : — 


32 

80 

89 

92 

2 

29 

66 

29 

6 

88 

98 

11 

48 

11 

1 

48  +  7 


B7 

=  351 

7 

5 

6 

80 

3 

1 

6 

5 

81 

1 

2 

66 

2 

09 

=  354 

9 

3 

5 

30 

7 

0 

98 

D  2  =  3  5  5  0  0!6  2964 

1|7  7  5  0  3 

4 


Take  E5 
From 


3550240471 
3550260182 


•0302496165  =  7  B 


=  -0039066973  =  90 


•0000868546  =  2  D 


•0000217146  =  5  E 


3550)  . 
F5 


G5 
H5 

12 
J3 


19711 
7750  =  -0000021715  =  5  F 

1961 

1775  =  -0000002172  =  5  G 


78  =  -0000000217  =  5  H 

8 

7  =  -0000000009  =  12 

II 

1  =  -0000000001  =  J  3 


Take    -0342672944 
From  3-5502601816 

Log.  3280-8992  =  3-5159928972 
.-.  log.  3-2808992  =  0-5159928972. 

The  constant  sidereal  year  consists  of  365-25636516  days ;  what 
is  the  log.  of  this  number  ? 

In  this  case  it  is  better  to  bring  the  constant  35502601816  to 
36525636516,  instead  of  bringing  the  given  number  to  the  con- 
stant, as  in  the  former  examples. 
2E 


350 


THE  PRACTICAL  MODEL  CALCULATOR. 

i 


B2 


35 

5 

7 

0 
1 

26 
00 
35 

01 
52 
50 

81 
03 
26 

36 

2 
2 

1 

8 

62 
97 
10 

0411 
2963 
1405 

•0086427476  =  2  B 


2028 


08  =  36506949|827  =  -0034726298  =  80 


825 


3|475 
3J651 


Take 
From 

D  5  =  365252 
365256 

06953 
36516 

36525-2)         4 
El  =         3 

29563 

65252 

Fl  = 
G7  = 

64311 
36525 

2j7  7  8  6 
2|5568 
2218 

H6 
10 

J7  = 


•0002171364  =  5  D 


•0000043429  =  1 E 


=  -0000004343  =  1  F 


•0000003040  =  7  G 


2191  =  -0000000261  =  6  H 


2,7 

25 


•0000000003  =  7  J 


•0123376214 
Add  3-5502601816 

Hence,  log.  3652-5636516  =  3-5625978030 
.-.     log.  365-25636516  =  2-562597803. 

M.  Regnault  determined  with  the  greatest  care  the  density  of 
mercury  to  be  13-59593  at  the  temperature  0°,  centigrade.     It  is 
required  to  calculate  the  log.  of  13-59593,  to  eight  places  of  decimals. 
In  this  case  it  is  better  to  bring  the  given  number  to  the  constant 
1371288574.       13595  9|3  0  0 
l|087674 
3807 


•003472630  =  80 


=  -000217136  =  5  D 


=  -000026058  =  E  6 


•000003909  =  F  9 


•000000022  =  H  5 


Subtract 
From 

08  =  1370 

5078 
6852 
1 

D5  =  1371 
1371 

1932 

2885 

E6  = 

952 

:     822 

130 
F9  =  123^ 

Jf 
H5  =        6| 

•003719755 


APPLICATION   OF   LOGARITHMS.  351 

Take  -003719755 
From  -137128857 

log.  1-359593  -  .-133409102 
.-.  log.  13-59593  =  1-133409102. 

TO  DETERMINE  THE  NUMBER  CORRESPONDING  TO  A  GIVEN  LOGARITHM. 

This  problem  has  been  very  much  neglected — so  much  so,  that 
none  of  our  elementary  books  ever  allude  to  a  method  of  comput- 
ing the  number  answering  to  a  given  logarithm.  When  an  opera- 
tion is  performed  by  the  use  of  logarithms,  it  is  very  seldom  that 
the  resulting  logarithm  can  be  found  in  the  table ;  we  have,  there- 
fore, to  find  the  nearest  less  logarithm,  and  the  next  greater,  and 
correct  them  by  proportion,  so  that  there  may  be  found  an  inter- 
mediate number  that  will  agree  with  the  given  logarithm,  or  nearly 
so.  But  although  the  proportional  parts  of  the  difference  abridge 
this  process,  we  can  only  find  a  number  appertaining  to  any  loga- 
rithm to  seven  places  of  figures  when  using  our  best  modern  tables. 
As,  however,  the  tabular  logarithms  extend  only  to  a  degree  of 
approximation,  fixed  generally  at  seven  decimal  places,  all  of  which^ 
except  those  answering  to  the  number  10  and  its  powers,  err,  either 
in  excess  or  defect,  the  maximum  limit  of  which  is  \  in  the  last 
decimal,  and  since  both  errors  may  conspire,  the  7th  figure  cannot 
be  depended  on  as  strictly  true,  unless  the  proposed  logarithm  falls 
between  the  limits  of  log.  10000  and  log.  22200. 

Indubitably  we  are  now  speaking  of  extreme  cases,  but  since  it 
is  not  an  unfrequent  occurrence  that  some  calculations  require  the 
most  rigid  accuracy,  and  many  resulting  logarithms  may  be  ex- 
tended beyond  the  limits  of  the  table,  this  subject  ought  to  have 
a  place  in  a  work  like  the  present.  It  is  not  part  of  the  present 
design  to  enter  into  a  strict  or  formal  demonstration  of  the  follow- 
ing mode  of  finding  the  number  corresponding  to  a  given  logarithm, 
as  the  operation  will  be  fully  explained  by  suitable  examples. 

What  number  corresponds  to  the  logarithm  3-4449555  ? 

The  next  less 'constant  log.  to  the  one  proposed  is  2-3758^209, 
or  rather,  3-37581209,  when  the  characteristic  or  index  is  increased 
by  a  unit.  Secondly. 

First  from  3-44496555  213  7  5  8  1  2  0  9  constant 

take  3-37581209  |23758121  =  A1 


•06915346  261339 


•04139269  =  1A         l|568036 


•02776077  - 0  „ 

•02592824  =  6  B 


3  3 


0 
0 
392009 


..183253 
173631  =40 

....9622 

8685  =  2  D 


277 


416 

109668 
664 


, .  937  2  7  8  5|2  8  2  9|8  =  C  4 


352 


THE  PRACTICAL  MODEL  CALCULATOR. 


93T 


68 
43 

25 

22 


IF 


5G 


3  =  7H 


27852 
If 

829 
570 

8  = 
6 
3 

04 

D2 
E  2 
Fl 
G5 
H7 

27858 

4 

5 

0 
5 

2 
1 

0 

7 
7 
3 
1 

7  = 
2  = 
9  = 
9  = 
9  = 

278590016 
.-.  2785-90016  is  the  number  sought. 
What  number  corresponds  to  the  logarithm  5-73437574  ? 
When  the  index  of  this  log.  is  reduced  by  a  unit,  the  nearest 
next  less  constant  is  4-66924683. 
From  4-73437574 
Take  4-66924683 
•6512891 

4139269 1  A 

•2373622" 

2160687 5  B 

•  •  212035 

173631 40 

. . .  39304 
39085, 


9D 

219     There  is  neither  the  equal  of 

•  217 5  F   this  number,  nor  a 

...0  G  less,  obtainable  from 


2...   ..4  TT  E, 

om 
Then,  466924683 
46692468  A3 

51 

2 

3617 

5680 
513 
5 

15 

85 
61 
13 

2 

1 

8 
7 
6 
6 

539816 
2159 
3 

788  B5 

267 
239 

2 

54 

197929 

48778 

1Q 
\y 

6  0  4 

1 
5   . 
2  D9 
2  F5 
2..  H4 

54 

2467 

2 

27 
71 

2 

542470006 
.-.  542470-006  is  the  number  whose  logarithm  is  5-73437574. 


THE    APPLICATION   OF   LOGARITHMS. 


353 


Had  the  given  logarithm  represented  a  decimal  with  a  positive 
index,  the  required  number  would  be  0-000054247,  &c. ;  or  if 
written  with  a  negative  index,  as  5-73437574,  the  result  would  be 
the  same,  for  the  characteristic  5,  shows  how  many  places  the  first 
significant  figure  is  below  unity. 

Required  the  number  corresponding  to  log.  2-3727451. 

The  constant  100000000  is  the  one  to  be  employed  in  this  case. 

1-3727451  the  given  log.  minus  1  in  the  index. 
1-0000000 

•3727451 

3725342 9  A 


.  .  .  2109 
1737.... 

4D 

....  372 
347  ... 

..    .8  E 

25 

22... 

,....5  F 

3 
3... 

...7  Gr 

10 

0 

0 

0 

010 

0  Constant. 

9 

0 

0 

0 

o|o 

0 

3 

6 

0 

0 

o 

0 

0 

8 

4 

0 

0 

0 

0 

1 

2 

6 

0 

0 

0 

1 

2 

6 

0 

0 

8 

4 

0 

3 

6 

23579485 
9432 

A9 

1 

D4 

E8 

23588|9 
18 

1 

9 

8 
7 

1 

1 

8 

F5 

1 

6 

G7 

23590949 

.*.  235-90949  is  the  required  number,  and  the  seconds  in  the  di- 
urnal apparent  motion  of  the  stars. 

235-90949"  =  3'  55-90949''. 

Let  it  be  required   to  find   the  hyperbolic  logarithm  of  any 

number,  as  3-1415926536.     The  common  log.  of  this  number  is 

•49714987269  (33),  and  the  common  log.  of  this  log.  isl-6964873. 

The  modulus  of  the  common  system  of  logarithms  is  -4342944819, 

&c. 


1  :  4342944819 

2  E2 


hyperbolic  log.  N :  common  log.  N. 

23 


354 


THE  PRACTICAL  MODEL  CALCULATOR. 


To  distinguish  the  hyperbolic  logarithm  of  the  number  N  from 
its  common  logarithm,  it  is  necessary  to  write  the  hyp.  log.  Log.  N, 
and  the  common  logarithm  log.  N. 

Hence,  4342944819  x  Log.  N  =  log.  N ; 
or  log.  (-4342944819)  +  log.  (log.  N)  =  log.  (log,  N). 
.-.  log.  (Log.  N)  =  log.  (log.  N)  -  1-6377843 ;  forT-6377843  = 
log.  -4342944819. 

Now,  to  work  the  above  example,  from  1-6964873 

take  T:6377843 

•0587030,  the    number 

corresponding  to  this  com.  log.  will  be  the  hyp.  log.  of  3-1415927. 
•0587030  must  be  reduced  to  -0000000  which  is  known  to  be  the 
log.  of  1. 

•0587030  1  A  =  1 1 0  0  0  0  0  010 

0413927   1A  4400000 


. 173103 

172855 

248 

217 

31 

30 

f     — — 


1  A 

4B 
5E 

7F 

2G 


66000 
1440 


1144664i4 

72 
80 


1  = 
3  = 
1  = 


3  =  G2 


114472988 

.-.  1-14472988  is  the  hyperbolic  log.  of  3-1415927,  true  to  the 
last  figure ;  for  the  hyp.  log.  3-1415926535898  =  1-1447298858494. 

The  reason  of  this  operation  is  very  clear,  because 
1  x  1-1  x  (1-01)4  x  (1-00001)5  x  (1-000001)7  x  (1-0000001)2  = 
1-14472988. 

This  example  answers  the  purpose  of  illustration,  but  the  hyp. 
log.  of  3-1415927  can  be  more  readily  found  by  dividing  its  com. 
log.  -49714987269  by  the  constant  -4342944819,  which  is  termed 
the  modulus  of  the  common  system  of  logarithms. 

Suppose  it  is  known  that  1-3426139  is  the  log.  of  the  decimal 
which  a  French  litre  is  of  an  English  gallon.  Required  the  decimal. 

The  index,  1,  may  be  changed  to  any  other  characteristic,  so  as 
to  suit  any  of  the  constants,  as  the  alteration  is  easily  allowed  for 
when  the  work  is  completed.  In  this  instance,  it  is  best  to  put 

+  1  instead  of  1. 

1 


8  A 


=  2B 


=  60 


From 
Take 

1-3426139 
1-0000000 

•3426139 
3311415 
•0114724 

.  .86427 

28297 
26045 

2252 


00 
80 

0;0 
00 

0 

0 

0 
0 

0(0  Constant 
00 

28 

00 

0 

0 

0 

0 

5 

6!0 

0 

0 

o 

0 

7iO 

0 

00 

0 

5 

6 

0 

0 

0 

2 

8 

0 

0 

8 

0 

1 

21|43|58|88|1  =  A  8 


THE   APPLICATION   OF  LOGARITHMS. 


355 


2252 
2171 


5D 


81 

43  =  1E 

38 

35 

3 


8F 


21|4 
4 

3 

2 

r 

2 

8 
7 
1 

8 
1 
4 

8 
7 
3 

1  = 

8 
6 

A8 
B2 

C6 

D  5 
E  1 
F8 
G7 

21 

-j 

6 
3 

6 
1 

7 
2 
3 

4 

0 

2 

95  = 
05 

80 
4 

21 

9 

9 
1 

8 
0 

278 
999 
2 

4  = 
1 

2 

22 

0 

0 

9 

2 
o 

1 

7 

? 

7 

97  = 

01  =- 
61  =» 
54  - 

220096913 
.-.  The  French  litre  =  -2200969  English  gallons. 
In  measuring  heights  by  the  barometer,  it  is  necessary  to  know 
the  ratio  of  the  density  of  the  mercury  to  that  of  the  air. 

At  Paris,  a  litre  of  air  at  0°  centigrade,  under  a  pressure  of  760 
millimetres,  weighs  1-293187  grammes.  At  the  level  of  the  sea, 
in  latitude  45°,  it  weighs  1-292697  grammes.  A  litre  of  water, 
at  its  maximum  density,  weighs  1000  grammes,  and  a  litre  of  mer- 
cury, at  the  temperature  of  0°  cent.,  weighs  13595-93  grammes : 

13595-93 

the  ratio  at  45° 


'  1-292697 

Now,  log.  13595-93  =  4-133409102 

and  log.  1-292697  =  0-111496744 


(29) 
(30) 


4-021912358  =  the  log.  of  the  ratio  at  45°. 

To  find  the  number  corresponding  to  this  log. ,  it  is  necessary  to  reject 
the  index  for  the  present,  and  reduce  the  Decimal  part  to  zero.  By 
this  means  the  necessity  of  using  any  of  the  constants  is  superseded. 

•021912358 

•021606869  =  5  B 


. . 305489 
303991 

....  1498 
1303 


7D 


3F 


195 

174  =  4G 

21 

17  =  4  H 

4 

4  =  91 

13595-93 

by  logarithms, 


1UOU 
'50 
1 

uu 

00 
00 
1 

U 

0 
0 
0 

u 
0 
0 
0 

I 

0 
0 

5 

U 
0 
0 
0 
0 

B 

D 

G 
II 
I 

5 

7 

4 
4 

9 

1051 

0 

7 

10 
35 

0 

7 
2 

>5 
1 

2 

= 

10517 

4 

5:9 
31 
4 

8 

6 
f) 

4 
1 

105174961 
=  10517-49,  &c.,  which  is  easily 


verified  by  common  division. 


356 


THE  PRACTICAL  MODEL  CALCULATOR. 


M.  Regnault  found  that,  at  Paris,  the  litre  of  atmospheric  air 
weighs  1-293187  grammes ;  the  litre  of  nitrogen  1*256167  grammes; 
a  litre  of  oxygen,  1-429802  grammes;  of  hydrogen,  0-089578 
grammes  ;  and  of  carbonic  acid,  1-977414  grammes.  But,  strictly 
considered,  these  numbers  are  only  correct  for  the  locality  in  which 
the  experiments  were  made ;  that  is  for  the  latitude  of  48°  50'  14" 
and  a  height  about  60  metres  above  the  level  of  the  sea ;  M.  Reg- 
nault finds  the  weight  of  the  litre  of  air  under  the  parallel  of  45° 
latitude,  and  at  the  same  distance  from  the  centre  of  the  earth  as 
that  which  the  experiments  were  tried,  to  be  12-926697. 

Assuming  this  as  the  standard,  he  deduces  for  any  other  latitude, 
any  other  distance  from  the  centre  of  the  earth,  the  formula, 

1-292697  (1-00001885)  (1  -  0-002837)  cos.  2  x 


w  — 


11 


Here,  w  is  the  weight  of  the  litre  of  air,  R  the  mean  radius  of 
the  earth  ==  6366198  metres,  h  the  height  of  the  place  of  observa- 
tion above  the  mean  radius,  and  ^  the  latitude  of  the  place. 

At  Philadelphia,  lat.  39°  56'  51-5",  suppose  the  radius  of  the 
earth  to  be  6367653  metres,  the  weight  of  the  litre  of  air  will  be 
1-2914892  grammes.  The  ratio  of  the  density  of  mercury  to  that 
of  air  at  the  level  of  the  sea  at  Philadelphia  is  10527*735  to  1 ; 
required  the  number  of  degrees  in  an  arc  whose  length  is  equal  to 
that  of  the  radius. 

360 
As  3-1415926535898  :  1  : :  -5-  :  the  required  degrees. 

Log.  360  =  2*556302500767 
log.  3*14159265359  =  0*497149872694 

2*059452623073 
log.  2  =  0*301029995664 

1*758122632409  =  the  log.  of  the* 
number  required. 

When  the  index  of  this  log.  is  changed  into  4,  the  nearest  next 
less  constant  is  4*669246832878. 

From  4*758122632409 
Take  4-669246832878 


2  A  = 
IB  = 
40  = 

7E  - 

•088875799531 
•82785370316 

.  .  6090429215 
4321373783 

..1769055432 
1736309917 

....32745515 
30400462 

..2345053 

4|6|6 
9'3 
4 

9 
3 

0 

2 
8 
6 

4 
4 
9 

6 

9 
2 

8 
3 
4 

3 

6 

2 
6 
8 

o 

5 
3 

7 
7 

9 

8  =  Constant 
6 
9 

56|49 
56 

7 
4 

88 
97 

6 

8 

67 

86 

78 
67 

3  =  A2 

8 

570 

2 

628 

282 

o 

655 
514 
423 

2 

446 
621 

771 
282 

1  =  B  1 

8 
9 
5 

6 


5  7  2  9  1|4  5  9  6  1|2  2  9'*=  C  4 


THE   APPLICATION    OP   LOGARITHMS. 


357 


5F  = 


3G  = 


9H 


91 


6  J 


8K 


2&45053 
2171471 

. 173582 
130288 

..43294 
39087 

...4207 
3909 


57291459611229 

4010401217 

12031 


04 


298 
261 

,737 
35 

.72 

2 


57295417 
2|8 

01347 

64773 
5 

7 
5 

7 

— 

E7 

F5 
G3 
H9 
I  9 
J  6 
K8 
L5 

5729575 

5 

5 

r> 

1 
1 

5 

1 
8 
5 
1 
3 

2 
8 
6 
5 
4 
4 

69  = 
73  = 
62  = 
66  = 

38  = 
58  = 
29  = 

5729577951295  =  the  num- 
5  L  =  2  ber  required. 

But  the  original  index  is  1;  .-.  57-29577951295°  are  the  num-  . 
ber  of  degrees  in  an  arc  the  length  of  which  is  equal  to  that  of  the 
radius. 

The  above  result  may  be  easily  verified  by  common  division,  a 
method,  no  doubt,  which  would  be  preferred  by  many,  for  loga- 
rithms are  seldom  used  when  the  ordinary  rules  of  arithmetic  can 
be  applied  with  any  reasonable  facility.  However,  this  example, 
like  many  others,  is  introduced  to  show  with  what  ease  and  correct- 
ness the  number  corresponding  to  a  given  log.  can  be  obtained. 
The.  extent,  also,  by  far  exceeds  that  obtainable  by  any  tables 
extant. 

Other  computations  give, 

r°  =  57-2957795130°  =  57°  17'  44"  -80624 
the  degrees  in  an  arc  =  radius. 

/  =  3437-7467707849'  =  3437'  44"  -80624 
fhe  minutes  in  an  arc  =  radius. 

r"  =  206264-8062470963 
the  number  of  seconds  in  an  arc  =  radius. 

The  relative  mean  motion  of  the  moon  from  the  sun  in  a  Julian  or 
fictitious  year,  of  365 J  days,  is  12  cir.4  signs,  12°  40' 15-977315'  = 
16029615-977315". 

.-.  16029615-977315"  :  1  circumference  (=  129600") 
: :  365-25  days 

:  29-5305889216  days  =  the  mean  synodic  month.  , 
This  proportion  may,  for  the  sake  of  example,  be  found  by  loga- 
rithms. 

Log.  365-25 2-56259022460634 

log.  1296000 6-11260500153457 

8-67519522614(391 
log.  16029615-977315  =  7-20492311805406 

1-47027210808685 


358 


THE    PRACTICAL   MODEL    CALCULATOR. 


If  the  index  of  this  log.  be  made  2  instead  of  1,  the  nearest  next 
less  constant  will  be  2-375812087593221. 


=  2A 


2B 


Fror 
Tab 

Q  2-47027210808685 
3  2-37581208759322 

2  A 

•09446002049363 
=  08278537031645 

2B 

.1167465017718 
=   864274756529 

60 

.  .  303190261189 
=   260446487591 

9D 

...42743773598 
=    39084549177 

8E 

....  3659224421 
=    3474338483 

4F 

.184885938 
=     173717706 

11168232 

2G 

=       8685889 

2482343 

5H 

=       2171473 
310870 

71 

=       304006 

6863 

U 

=         4343 

2520 

5K 

=         2172 
348 

8L 

2N 

=          347 
=           1 

^  o 

<  <± 

57 

(  0 

49 

28 

^  u 
46 

74 

z  o  y  o 
52:51 

73J26 

I  13 

97|6 
260 

293 
1 

2 
7 

51 

59 
4 

475 

508 
398 
5 

177 

851 
772 
865 
4 

015 
062 

128 
029 
399 

2950 

1538 

8669 

635 

2 

6551 

3849 

803 

10 

6205 

540 

24 

781 

4 

=  C6 


29 

528 
2 

1 

3 

0087 
6224 

8 

4976 
8070 
2678 
1 

295304 
1 

63 

18 

20 
12 

57 
18 
1 

26 
52 

77 

29 

5 

3 

058 

13 

59 

277 
061 

56 
16 

=  F4 


29 

530587 
1 

2 
4 

313 

76 

8173 
529 

3 
4 

I 

G2 
H5 

2 

0 

6 

714 

1 

= 

17 

2 

953 

1 

= 

Jl 

1 

4i76 

5 

—  - 

K5 

236 

2 

=: 

L8 

6 

= 

N2 

295305889217832 
.;.  29-5305889218  is  the  number  required. 
To  perform,  by  logarithms,  the  ordinary  operations  of  multipli- 
cation, division,  proportion,  or  even  the  extraction  of  the  square 
root,  except  in  the  way  of  illustration,  is  not  the  design  of  these 
pages ;  for  such  an  application  of  logarithms,  in  a  particular  man- 
ner only,  diminish  the  labour  of  the  operator.     It  is  not  necessary, 
however,  to  examine  minutely  here  the  instances  in  which  common 
arithmetic  is  preferable  to  artificial  numbers ;  besides,  much  \vili 
depend  on  the  skill  and  facility  of  the  operator. 


359 


TRIGONOMETRY. 

ANGULAR    MAGNITUDES. — TRIGONOMETRY. — HEIGHT    AND    DISTANCES. — 

SPHERICAL    TRIGONOMETRY. THE    APPLICATION    OF    LOGARITHMS     TO 

ANGULAR    MAGNITUDES. 

PLANE  TRIGONOMETRY  treats  of  the  relations  and  calculations  of 
the  sides  and  angles  of  plane  triangles. 

The  circumference  of  every  circle  is  supposed  to  be  divided  into 
360  equal  parts,  called  degrees ;  also  each  degree  into  60  minutes, 
each  minute  into  60  seconds,  and  so  on. 

Hence  a  semicircle  contains  180  degrees,  and  a  quadrant  90  de- 
grees. 

The  measure  of  any  angle  is  an  arc  of  any  circle  contained  be- 
tween the  two  lines  which  form  that  angle,  the  angular  point  being 
the  centre ;  and  it  is  estimated  by  the  number  of  degrees  contained 
in  that  arc. 

Hence,  a  right  angle  being  measured  by  a  quadrant,  or  quarter 
of  the  circle,  is  an  angle  of  90  degrees ;  and  the  sum  of  the  three 
angles  of  every  triangle,  or  two  right  angles,  is  equal  to  180  de- 
grees. Therefore,  in  a  right-angled  triangle,  taking  one  of  the 
acute  angles  from  90  degrees,  leaves  the  other  acute  angle  ;  and 
the  sum  of  two  angles,  in  any  triangle,  taken  from  180  degrees, 
leaves  the  third  angle ;  or  one  angle  being  taken  from  180  degrees, 
leaves  the  sum  of  the  other  two  angles. 

Degrees  are  marked  at  the  top  of  the  figure  with  a  small  °,  mi- 
nutes with  ',  seconds  with  ",  and  so  on.  Thus,  57°  30'  12"  de- 
note 57  degrees  30  minutes  and  12  seconds. 

The  complement  of  an  arc,  is  what  it  wants  of 
a  quadrant  or  90°.  Thus,  if  AD  be  a  quadrant, 
then  BD  is  the  complement  of  the  arc  AB  ;  and, 
'reciprocally,  AB  is  the  complement  of  BD.  So  E 
that,  if  AB  be  an  arc  of  50°,  then  its  complement 
BD-will  be  40°. 

The  supplement  of  an  arc,  is  what  it  wants  of 
a  semicircle,  or  180°.  Thus,  if  ADE  be  a  semicircle,  then  BDE 
is  the  supplement  of  the  arc  AB  ;  and,  reciprocally,  AB  is  the  sup- 
plement of  the  arc  BDE.  So  that,  if  AB  be  an  arc  of  50°,  then 
its  supplement  BDE  will  be  130°. 

The  sine,  or  right  sine,  of  an  arc,  is  the  line  drawn  from  one 
extremity  of  the  arc,  perpendicular  to  the  diameter  passing  through 
the  other  extremity.  Thus,  BF  is  the  sine  of  the  arc  AB,  or  of 
the  arc  BDE. 

Hence  the  sine  (BF)  is  half  the  chord  (BG)  of  the  double  arc 
(BAG). 

The  versed  sine  of  an  arc,  is  the  part  of  the  diameter  intercepted 
between  the  arc  and  its  sine.  So,  AF  is  the  versed  sine  of  the  arc 
AB,  and  EF  the  versed  sine  of  the  arc  EDB. 


360 


THE  PRACTICAL  MODEL  CALCULATOR. 


The  tangent  of  an  arc  is  a  line  touching  the  circle  in  one  ex- 
tremity of  that  arc,  continued  from  thence  to  meet  a  line  drawn 
from  the  centre  through  the  other  extremity :  which  last  line  is 
called  the  secant  of  the  same  arc.  Thus,  AH  is  the  tangent,  and 
CH  the  secant,  of  the  arc  AB.  Also,  El  is  the  tangent,  and  CI 
the  secant,  of  the  supplemental  arc  BDE.  And  this  latter  tangent 
and  secant  are  equal  to  the  former,  but  are  accounted  negative,  as 
being  drawn  in  an  opposite  or  contrary  direction  to  the  former. 

The  cosine,  cotangent,  and  cosecant,  of  an  arc,  are  the  sine, 
tangent,  and  secant  of  the  complement  of  that  arc,  the  co  being 
only  a  contraction  of  the  word  complement.  Thus,  the  arcs  AB, 
BD  being  the  complements  of  each  other,  the  sine,  tangent  or  se- 
cant of  the  one  of  these,  is  the  cosine,  cotangent  or  cosecant  of  the 
other.  So,  BF,  the  sine  of  AB,  is  the  cosine  of  BD ;  and  BK, 
the  sine  of  BD,  is  the  cosine  of  AB :  in  like  manner,  AH,  the 
tangent  of  AB,  is  the  cotangent  of  BD ;  and  DL,  the  tangent  of 
DB,  is  the  cotangent  of  AB  :  also,  CH,  the  secant  of  AB,  is  the 
cosecant  of  BD  ;  and  CL,  the  secant  of  BD,  is  the  cosecant  of  AB. 

Hence  several  remarkable  properties  easily  follow  from  these 
definitions ;  as, 

That  an  arc  and  its  supplement  have  the  same  sine,  tangent,  and 
secant ;  but  the  two  latter,  the  tangent  and  secant,  are  accounted 
negative  when  the  arc  is  greater  than  a  quadrant  or  90  degrees. 

When  the  arc  is  0,  or  nothing,  the  sine  and  tangent  are  nothing, 
but  the  secant  is  then  the  radius  CA.  But  when  the  arc  is  a 
quadrant  AD,  then  the  sine  is  the  greatest  it  can  be,  being  the  ra- 
dius CD  of  the  circle  ;  and  both  the  tangent  and  secant  are  infinite. 

Of  any  arc  AB,  the  versed  sine  AF, 
and  cosine  BK,  or  CF,  together  make 
up  the  radius  CA  of  the  circle.  The 
radius  CA,  tangent  AH,  and  secant 
CH,  form  a  right-angled  triangle  CAH. 
So  also  do  the  radius,  sine,  and  cosine, 
form  another  right-angled  triangle 
CBF  or  CBK.  As  also  the  radius, 
cotangent,  and  cosecant,  another  right- 
angled  triangle  CDL.  And  all  these 
right-angled  triangles  are  similar  to 
each  other. 

The  sine,  tangent,  or  secant  of  an 
angle,  is  the  sine,  tangent,  or  secant 
of  the  arc  by  which  the  angle  is  mea- 
sured, or  of  the  degrees,  &c.  in  the  same 
arc  or  angle. 

The  method  of  constructing  the  scales 
of  chords,  sines,  tangents,  and  secants, 
usually  engraven  on  instruments,  for 
practice,  is  exhibited  in  the  annexed 
figure. 


75 


TRIGONOMETRY.  361 

A  trigonometrical  canon,  is  a  table  exhibiting  the  length  of  the 
sine,  tangent,  and  secant,  to  every  degree  and  minute  of  the  quad- 
rant, with  respect  to  the  radius,  which  is  expressed  by  unity,  or  1, 
and  conceived  to  be  divided  into  10000000  or  more  decimal  parts. 
And  further,  the  logarithms  of  these  sines,  tangents,  and  secants 
are  also  ranged  in  the  tables  ;  which  are  most  commonly  used,  as 
they  perform  the  calculations  by  only  addition  and  subtraction, 
instead  of  the  multiplication  and  division  by  the  natural  sines,  &c., 
according  to  the  nature  of  logarithms. 

Upon  this  table  depends  the  numeral  solution  of  the  several 
cases  in  trigonometry.  It  will  therefore  be  proper  to  begin  with 
the  mode  of  constructing  it,  which  may  be  done  in  the  following 
manner  :  — 

To  find  the  sine  and  cosine  of  a  given  arc. 

This  problem  is  resolved  after  various  ways.  One  of  these  is  as 
follows,  viz.  by  means  of  the  ratio  between  the  diameter  and  cir- 
cumference of  a  circle,  together  with  the  known  series  for  the  sine 
and  cosine,  hereafter  demonstrated.  Thus,  the  semi-circumference 
of  the  circle,  whose  radius  is  1,  being  3-141592653589793,  &c., 
the  'proportion  will  therefore  be, 

As  the  number  of  degrees  or  minutes  in  the  semicircle, 
Is  to  the  degrees  or  minutes  in  the  proposed  arc, 
So  is  3-14159265,  &c.,  to  the  length  of  the  said  arc. 
This  length  of  the  arc  being  denoted  by  the  letter  a;  also  its 
sine  and  cosine  by  s  and  c  ;  then  will  these  two  be  expressed  by  the 
two  following  series,  viz.  :  — 


~ 


2.3.4.5.6.7 


a 


120       5Q40  < 

a?          a4  a6 

'  "2  +  2^4  ~"  2.3.4.5.6  +  &c< 

a2       a4         a6 
~  2  +  24  "  720  +  &c' 

If  it  be  required  to  find  the  sine  and  cosine  of  one  minute. 
Then,  the  number  of  minutes  in  180°  being  10800,  it  will  be  first, 
as  10800  :  1  :  :  3-14159265,  &c.  :  -000290888208665  =  the  length 
of  an  arc  of  one  minute.  Therefore,  in  this  case, 

a  =    -0002908882 
and  i#3  =    -000000000004,  &c. 

the  difference  is  s  =     -0002908882  the  sine  of  1  minute. 
Also,  from       1- 

take  |a2  =  0-0000000423079,  &c. 

leaves  c  =    -9999999577  the  cosine  of  1  minute. 
2  F 


362  THE    PRACTICAL   MODEL    CALCULATOR. 

For  the  sine  and  cosine  of  5  degrees. 

Here,  as  180°  :  5°  :  :  3-14159265,  &c.,  :  -08726646  =  a  the 
length  of  5  degrees. 

Hence,  (t  =  -08726646 

_  1^3  ^  _  .Q0011076 
=  -00000004 


these  collected  give  *  =  -08715574  the  sine  of  5°. 

And,  for  the  cosine,         1  =    1- 

-  jaa  =  _  -00380771  ' 
+  Aa4  =       -00000241 

these  collected,  give  c  =  -99619470  the  consine  of  5°. 

After  the  same  manner,  the  sine  and  cosine  of  any  other  arc 
may  be  computed.  But  the  greater  the  arc  is,  the  slower  the  series 
will  converge,  in  which  case  a  greater  number  of  terms  must  be 
taken  to  bring  out  the  conclusion  to  the  same  degree  of  exactness. 

Or,  having  found  the  sine,  the  cosine  will  be  found  from  it,  by 
the  property  of  the  right-angled  triangle  CBF,  viz.  the  cosine 
OF  =  N/CB2-  BF2,  or  c  =  v/1  -  s2. 

There  are  also  other  methods  of  constructing  the  canon  of  smes 
and  cosines,  which,  for  brevity's  sake,  are  here  omitted. 

To  compute  the  tangents  and  secants. 

The  sines  and  cosines  being  known,  or  found,  by  the  foregoing 
problem;  the  tangents  and  secants  will  be  easily  found,  from  the 
principle  of  similar  triangles,  in  the  following  manner  :  — 

In  the  first  figure,  where,  of  the  arc  AB,  BF  is  the  sine,  OF  or 
BK  the  cosine,  AH  the  tangent,  CH  the  secant,  DL  the  cotangent, 
and  CL  the  cosecant,  the  radius  being  CA,  or  CB,  or  CD  ;  the 
three  similar  triangles  CFB,  CAH,  CDL,  give  the  following  pro- 
portions : 

1.  CF  :  FB  :  :  CA  :  AH  ;  whence  the  tangent  is  known,  being 
a  fourth  proportional  to  the  cosine,  sine,  and  radius. 

2.  CF  :  CB  :  :  CA  :  CH  ;  whence  the  secant  is  known,  being  a 
third  proportional  to  the  cosine  and  radius. 

3.  BF  :  FC  :  :  CD  :  DL  ;  whence  the  cotangent  is  known,  being 
a  fourth  proportional  to  the  sine,  cosine,  and  radius. 

4.  BF  :  BC  :  :  CD  :  CL;  whence  the  cosecant  is  known,  being 
a  third  proportional  to  the  sine  and  radius. 

Having  given  an  idea  of  the  calculations  of  sines,  tangents,  and 
secants,  we  may  now  proceed  to  resolve  the  several  cases  of  trigo- 
nometry ;  previous  to  which,  however,  it  may  be  proper  to  add  a 
few  preparatory  notes  and  observations,  as  below. 

There  are  usually  three  methods  of  resolving  triangles,  or  the 
cases  of  trigonometry  —  namely,  geometrical  construction,  arith* 
metical  computation,  and  instrumental  operation, 

In  the  first  method.  —  The  triangle  is  constructed  by  making  the 
parts  of  the  given  magnitudes,  namely,  the  sides  from  a  scale  of 


TRIGONOMETRY.  363 

equal  parts,  and  the  angles  from  a  scale  of  chords,  or  by  some 
other  instrument.  Then,  measuring  the  unknown  parts  by  the 
same  scales  or  instruments,  the  solution  will  be  obtained  near 
the  truth. 

In  the  second  method. — Having  stated  the  terms  of  the  propor- 
tion according  to  the  proper  rule  or  theorem,  resolve  it  like  any 
other  proportion,  in  which  a  fourth  term  is  to  be  found  from  three 
given  terms,  by  multiplying  the  second  and  third  together,  and 
dividing  the  product  by  the  first,  in  working  with  the  natural  num- 
bers ;  or,  in  working  with  the  logarithms,  add  the  logs,  of  the 
second  and  third  terms  together,  and  from  the  sum  take  the  log. 
of  the  first  term ;  then  the  natural  number  answering  to  the  re- 
mainder is  the  fourth  term  sought. 

In  the  third  method. — Or  instrumentally,  as  suppose  by  the  log. 
lines  on  one  side  of  the  common  two-foot  scales ;  extend  the  com- 
passes from  the  first  term  to  the  second  or  third,  which  happens  to 
be  of  the  same  kind  with  it ;  then  that  extent  will  reach  from  the 
other  term  to  the  fourth  term,  as  required,  taking  both  extents 
towards  the  same  end  of  the  scale. 

In  every  triangle,  or  case  in  trigonometry,  there  must  be  given 
three  parts,  to  find  the  other  three.  And,  of  the  three  parts  that 
are  given,  one  of  them  at  least  must  be  a  side ;  because  the  same 
angles  are  common  to  an  infinite  number  of  triangles. 

All  the  cases  in  trigonometry  may  be  comprised  in  three  vari- 
eties only ;  viz. 

1.  When  a  side  and  its  opposite  angle  are  given. 

2.  When  two  sides  and  the  contained  angle  are  given. 

3.  When  the  three  sides  are  given. 

For  there  cannot  possibly  be  more  than  these  three  varieties  of 
cases  ;  for  each  of  which  it  will  therefore  be  proper  to  give  a  sepa- 
rate theorem,  as  follows : 

When  a  side  and  its  opposite  angle  are  two  of  the  given  parts. 
Then  the  sides  of  the  triangle  have  the  same  proportion  to  each 
other,  as  the  sines  of  their  opposite  angles  have. 
That  is, 

As  any  one  side, 

Is  to  the  sine  of  its  opposite  angle ; 
So  is  any  other  side, 
To  the  sine  of  its  opposite  angle. 
For,  let  ABC  be  the  proposed  triangle,  having 
AB  the  greatest  side,  and  BC  the  least.     Take 

AD  =  BC,  considering  it  as  a  radius ;  and  let  

fall  the  perpendiculars  DE,  CF,  which  will  evi-  A  E  F  B 
dently  be  the  sines  of  the  angles  A  and  B,  to  the  radius  AD  or 
BC.  But  the  triangles  ADE,  ACF,  are  equiangular,  and  there- 
fore AC  :  CF  : :  AD  or  BC  :  DE ;  that  is,  AC  is  to  the  sine  of  its 
opposite  angle  B,  as  BC  to  the  sine  of  its  opposite  angle  A. 

In  practice,  to  find  an  angle,  begin  the  proportion  with  a  side 


364          THE  PRACTICAL  MODEL  CALCULATOR. 

opposite  a  given  angle.     And  to  find  a  side,  begin  with  an  angle 
opposite  a  given  side. 

An  angle  found  by  this  rule  is  ambiguous,  or  uncertain  whether 
it  be  acute  or  obtuse,  unless  it  be  a  right  angle,  or  unless  its  mag- 
nitude be  such  as  to  prevent  the  ambiguity ;  because  the  sine  an- 
swers to  two  angles,  which  are  supplements  to  each  other ;  and 
accordingly  the  geometrical  construction  forms  two  triangles  with 
the  same  parts  that  are  given,  as  in  the  example  below;  and  when 
there  is  no  restriction  or  limitation  included  in  the  question,  either 
of  them  may  be  taken.  The  degrees  in  the  table,  answering  to  the 
sine,  are  the  acute  angle  ;  but  if  the  angle  be  obtuse,  subtract  those 
degrees  from  180°,  and  the  remainder  will  be  the  obtuse  angle. 
When  a  given  angle  is  obtuse,  or  a  right  one,  there  can  be  no  am- 
biguity ;  for  then  neither  of  the  other  angles  can  be  obtuse,  and 
the  geometrical  construction  will  form  only  one  triangle. 

In  the  plane  triangle  ABC, 

f  AB  345  yards 
Given,  I  BC  232  yards 

(angle  A  37°  20' 
Required  the  other  parts. 

Geometrically. — Draw  an  indefinite  line,  upon  which  set  off  AB 
=  345,  from  some  convenient  scale  of  equal  parts.  Mabe  the 
angle  A  =  37J°.  With  a  radius  of  232,  taken  from  the  same 
scale  of  equal  parts,  and  centre  B,  cross  AC  in  the  two  points  C,  C. 
Lastly,  join  BC,  BC,  and  the  figure  is  constructed,  which  gives 
two  triangles,  showing  that  the  case  is  ambiguous. 

Then,  the  sides  AC  measured  by  the  scale  of  equal  parts,  and 
the  angles  B  and  C  measured  by  the  line  of  chords,  or  other  in- 
strument, will  be  found  to  be  nearly  as  below ;  viz. 

AC  174  angle  B  27°  angle  C  115|° 

or     374J  or         78i  or          64J 

Arithmetically. — First,  to  find  the  angles  at  C : 

As  side  BC  232 log.  2-3654880 

To  sin.  opp.  angle  A       37°  20' 9-7827958 

So  side  AB  345 2-5378191 

To  sin.  opp.  angle  C     115°  36'  or  64°  24 9-9551269 

Add  angle  A       37    20       37    20 

The  sum  152    56  or  101    44 

Taken  from       180    00      180    00 

Leaves  angle  B  27    04  or    78    16 

Then,  to  find  the  side  AC : 

As  sine  angle      A       37°  20'-. log.  9-7827958 

To  opposite  side  BC          232 2-365488 

a      •  i         -D    /27°04' 9-6580371 

So  sine  angle       B    j  7g    16 9-9908291 

To  opposite  side  AC     174-07 2-2407293 

or,      374-56 2-5735213 


TRIGONOMETRY. 


365 


In  the  plane  triangle  ABC, 
f  AB  365  poles 

Given,  1  angle  A       57°  12' 
(angle  B       24    45 
Required  the  other  parts. 

In  the  plane  triangle  ABC, 

f  AC  120  feet 

Given  J  BC  112  feet 

(angle  A      57°  27' 
Required  the  other  parts. 


Ans.  (angle  C  98°  3' 
AC  154-33 
BC  309-86 


Ans. 


angle  B  64°  34'  21" 

or,     115    25  39 

angle  C  57    58  39 

or,         7      7  21 

AB  112-65  feet 
L    or,          16-47  feet 


When  two  sides  and  their  contained  angle  are  given. 
Then  it  will  be, 

As  the  sum  of  those  two  sides,, 

Is  to  the  difference  of  the  same  sides ; 

So  is  the  tang,  of  half  the  sum  of  their  opposite  angles, 

To  the  tang,  of  half  the  difference  of  the  same  angles. 

Hence,  because  it  is  known  that  the  half  sum  of  any  two  quan- 
tities increased  by  their  half  difference,  gives  the  greater,  and  di- 
minished by  it  gives  the  less,  if  the  half  difference  of  the  angles, 
so  found,  be  added  to  their  half  sum,  it  will  give  the  greater  angle, 
and  subtracting  it  will  leave  the  less  angle. 

Then,  all  the  angles  being  now  known,  the  unknown  side  will  be 
found  by  the  former  theorem. 

Let  ABC  be  the  proposed  triangle,  having  E 
the  two  given  sides  AC,  BC,  including  the  given 
angle  C.  With  the  centre  C,  and  radius  CA, 
the  less  of  these  two  sides,  describe  a  semicircle, 
meeting  the  other  side  BC  produced  in  D  and  E. 
Join  AE,  AD,  and  draw  DF  parallel  to  AE.  F  B 

Then,  BE  is  the  sum,  and  BD  the  difference  of  the  two  given 
sides  CB,  CA.  Also,  the  sum  of  the  two  angles  CAB,  CBA,  is 
equal  to  the  sum  of  the  two  CAD,  CDA,  these  sums  being  each- 
the  supplement  of  the  vertical  angle  C  to  two  right  angles :  but 
the  two  latter  CAD,  CDA,  are  equal  to  each  other,  being  opposite  to 
the  two  equal  sides  CA,  CD  :  hence,  either  of  them,  as  CDA,  is  equal 
to  half  the  sum  of  the  two  unknown  angles  pAB,  CBA.  Again, 
the  exterior  angle  CDA  is  equal  to  the  two  interior  angles  B  and 
DAB  ;  therefore,  the  angle  DAB  is  equal  to  the  difference  between 
CDA  and  B,  or  between  CAD  and  B ;  consequently,  the  same 
angle  DAB  is  equal  to  half  the  difference  of  the  unknown  angles 
B  and  CAB  ;  of  which  it  has  been  shown  that  CDA  is  the  half  sum. 

Now  the  angle  DAE,  in  a  semicircle,  is  a  right  angle,  or  AE  is 
perpendicular  to  AD  ;  and  DF.  parallel  to  AE,  is  also  perpendicular 
2  F2 


366  THE   PRACTICAL    MODEL   CALCULATOR. 

to  AD :  consequently,  AE  is  the  tangent  of  CD  A  the  half  sum 
and  DF  the  tangent  of  DAB  the  half  difference  of  the  angles,  to 
the  same  radius  AD,  by  the  definition  of  a  tangent.  But,  the  tan- 
gents AE,  DF,  being  parallel,  it  will  be  as  BE  :  BD  : :  AE  :  DF ; 
that  is,  as  the  sum  of  the  sides  is  to  the  difference  of  the  sides,  so 
is  the  tangent  of  half  the  sum  of  the  opposite  angles,  to  the  tan- 
gent of  half  their  difference. 

The  sum  of  the  unknown  angles  is  found,  by  taking  the  given 
angle  from  180°. 

In  the  plane  triangle  ABC, 

(  AB  345  yards 

Given,  4  AC  174-07  yards 

(angle  A     37°  20' 

Required  the  other  parts. 

Geometrically. — Draw  AB  =  345  from  a  scale  of  equal  parts. 
Make  the  angle  A  =  37°  20'.  Set  off  AC  =  174  by  the  scale  of 
equal  parts.  Join  BC,  and  it  is  done. 

Then  the  other  parts  being  measured,  they  are  found  to  be  nearly 
as  follows,  viz.  the  side  BC  232  yards,  the  angle  B  27°,  and  the 
angle  C  1151°. 

Arithmetically. 

As  sum  of  sides  AB,  AC 519-07  log.    2-7152259 

To  difference  of  sides  AB,  AC 170-93  2-2328183 

So  tangent  half  sum  angles  C  and  B 71°  20'        10-4712979 

To  tangent  half  difference  angles  C  and  B  44    16  9-9888903 

Their  sum  gives  angle  C    115    36 
Their  diff.  gives  angle  B      27      4 

Then,  by  the  former  theorem, 

As  sine  angle  C  115°  36',  or  64°  24' log.  9-0551259 

To  its  opposite  side  AB  345 2-5378191 

So  sine  angle  A  37°  20' 9-7827958 

To  its  opposite  side  BC  232 2-3654890 

In  the  plane  triangle  ABC, 

{AB  365  poles 
AC  154-33 
angle  A     57°  12'  (        BC   309-86 

Required  the  other  parts.  <  angle  B  24°  45' 

( angle  C  98°    3' 

In  the  plane  triangle  ABC, 

(         AC  120  yards 
Given,  J          BC  112  yards 

(  angle  C  57°  58'  39"  f         AB  112-65 

Required  the  other  parts.  <  angle  A  57°  27'   0" 

(angle  B  64    34  21 


TRIGONOMETRY.  367 

When  the  three  sides  of  the  triangle  are  given. 

Then,  having  let  fall  a  perpendicular  from  the  greatest  angle 
upon  the  opposite  side,  or  base,  dividing  it  into  two  segments,  and 
the  whole  triangle  into  two  right-angled  triangles ;  it  will  be, 

As  the  base,  or  sum  of  the  segments, 

Is  to  the  sum  of  the  other  two  sides ; 

So  is  the  difference  of  those  sides, 

To  the  difference  of  the  segments  of  the  base. 

Then,  half  the  difference  of  the  segments  being  added  to  the 
half  sum,  or  the  half  base,  gives  the  greater  segment ;  and  the 
same  subtracted  gives  the  less  segment. 

Hence,  in  each  of  the  two  right-angled  triangles,  there  will  be 
known  two  sides,  and  the  angle  opposite  to  one  of  them ;  conse- 
quently, the  other  angles  will  be  found  by  the  first  problem. 

The  rectangle  under  the  sum  arid  difference  of  the  two  sides,  is 
equal  to  the  rectangle  under  the  sum  and  difference  of  the  two  seg- 
ments. Therefore,  by  forming  the  sides  of  these  rectangles  into 
a  proportion,  it  will  appear  that  the  sums  and  differences  are  pro- 
portional, as  in  this  theorem. 

In  the  plane  triangle  ABC,  c 

f  AB  345  yards 
Given,  the  sides  1  AC  232 

(BC  174-07 
To  find  the  angles. 

Geometrically. — Draw  the  base  AB  =  345  by  a  scale  of  equal 
parts.  With  radius  232,  and  centre  A,  describe  an  arc  ;  and  with 
radius  174,  and  centre  B,  describe  another  arc,  cutting  the  former 
in  C.  Join  AC,  BC,  and  it  is  done. 

Then,  by  measuring  the  angles,  they  will  be  found  to  be  nearly 
as  follows,  viz.  angle  A  27°,  angle  B  37 J°,  and  angle  C  115 J°. 

Arithmetically. — Having  let  fall  the  perpendicular  CP,  it  will  be, 
As  the  base  AB  :  AC  -f  BC  : :  AC  -  BC  :  AP  -  BP 
that  is,  as  345  :  406-07  : :  57-93  :  68-18  =  AP  —  BP 

its  half  is 34-09 

the  half  base  is 172-50 

the  sum  of  these  is 206-59  =  AP 

and  their  difference 138-41  =  BP 

Then,  in  the  triangle  APC,  right-angled  at  P, 

As  the  side  AC 232         log.  2-3654880 

To  sine  opposite  angle 90°       10-0000000 

So  is  side  AP 206-59    2-3151093 

To  sine  opposite  angle  ACP 62°  56' 9-9496213 

Which  taken  from 90    00 

Leaves  the  angle  A 27    04 


368  THE    PRACTICAL   MODEL   CALCULATOR. 

Again,  in  the  triangle  BPC,  right-angled  at  P, 

As  the  side  of  BC 174-07 log.  2-2407239 

To  sine  opposite  angle  P.. .  90°       10-0000000 

So  is  side  BP 138-41 2-1411675 

To  sin.  opposite  angle  BCP  52°  40' 9-9004436 

Which  taken  from 90    00 

Leaves  the  angle  B...  37    20 

Also,  the  angle  AGP...  62°  56' 

Added  to  angle  BCP...  52    40 

Gives  the  whole  angle  ACB...115    36 

So  that  all  the  three  angles  are  as  follow,  viz. 
the  angle  A  27°  4';    the  angle  B  37°  20';  the  angle  C  115°  36'. 
In  the  plane  triangle  ABC, 

f  AB  365  poles 
Given  the  sides,  1  AC  154-33 

(BC  309-86  (angle  A  57°  12' 

To  find  the  angles.  1  angle  B  24    45 

UngleC98      3 
In  the  plane  triangle  ABC, 

(AB120 
Given  the  sides  J  AC  112-65 

(  BC  112  f  angle  A  57°  27'  00" 

To  find  the  angles.  <  angle  B  57    58  39 

(  angle  C  64   34  21 

The  three  foregoing  theorems  include  all  the  cases  of  plane  tri- 
angles, both  right-angled  and  oblique ;  besides  which,  there  are 
other  theorems  suited  to  some  particular  forms  of  triangles,  which 
are  sometimes  more  expeditious  in  their  use  than  the  general  ones  ; 
one  of  which,  as  the  case  for  which  it  serves  so  frequently  occurs, 
may  be  here  taken,  as  follows  : — 

When,  in  a  right-angled  triangle,  there  are  given  one  leg  and  the 
angles  ;  to  find  the  other  leg  or  the  hypothenuse  ;  it  will  be, 

As  radius,  i.  e.  sine  of  90°  or  tangent  of  45° 

Is  to  the  given  leg, 

So  is  the  tangent  of  its  adjacent  angle 

To  the  other  leg ; 

And  so  is  the  secant  of  the  same  angle 

To  the  hypothenuse. 

AB  being  the  given  leg,  in  the  right-angled  tri- 
angle ABC  ;  with  the  centre  A,  and  any  assumed  ra- 
dius, AD,  describe  an  arc  DE,  and  draw  DF  perpen- 
dicular to  AB,  or  parallel  to  BC.  Now  it  is  evident, 
from  the  definitions,  that  DF  is  the  tangent,  and  AF 
the  secant,  of  the  arc  DE,  or  of  the  angle  A  which  A  v  0 
is  measured  by  that  arc,  to  the  radius  AD.  Then,  because  of  the 
parallels  BC,  DF,  it  will  be  as  AD  :  AB  : :  DF  :  BC  : :  AF  :  AC, 
which  is  the  same  as  the  theorem  is  in  words. 


OF   HEIGHTS   AND   DISTANCES.  369 

In  the  right-angled  triangle  ABC, 

to  find  AC  and  BC- 


Geometrically.  —  Make  AB  =  162  equal  parts,  and  the  angle  A  = 
53°  7'  48";  then  raise  the  perpendicular  BC,  meeting  AC  in  C. 
So  shall  AC  measure  270,  and  BC  216. 

Arithmetically. 
As  radius  .....................  tang.  45°  .........  log.  10-0000000 

TolegAB  ....................       162       .........    '       2-2095150 

So  tang,  angle  A  ............  53°  7'  48"  .........         10-1249371 

To  leg  BC  ....................       216       .........  2-3344521 

So  secant  angle  A  ...........  53°  7'  48"  .........         10-2218477 

To  hyp.  AC  ..................      270       .........  2-4313627 

In  the  right-angled  triangle  ABC, 
r.        /     the  leg  AB  180 
Jlven  t  the  angle  A  62°  40'  f  AQ  392.0147 

m  •    j?      i   J.1  ^i  •  i  7    £*\J   OV6  \J  J.T:  i 

To  find  the  other  two  sides.  <  T>Q  348-2464 

There  is  sometimes  given  another  method  for  right-angled  tri- 
angles, which  is  this  :  c 

ABC  being  such  a  triangle,  make  one  leg  AB  ra- 
dius, that  is,  with  centre  A,  and  distance  AB,  de- 
scribe an  arc  BF.  Then  it  is  evident  that  the  other 
leg  BC  represents  the  tangent,  and  the  hypothenuse 
AC  the  secant,  of  the  arc  BF,  or  of  the  angle  A. 

In  like  manner,  if  the  leg  BC  be  made  radius  ; 


then  the  other  leg  AB  will  represent  the  tangent,  and  tb.Q 
thenuse  AC  the  secant,  of  the  arc  BG  or  angle  C. 

But  if  the  hypothenuse  be  made  radius  ;  then  each  leg;  will,  re- 
present the  sine  of  its  opposite  angle  ;  namely,  the  leg  AB  the  sine 
of  the  arc  AE  or  angle  C,  and  the  leg  BC  the  sine  of  the  arc,  Cl> 
or  angle  A. 

And  then  the  general  rule  for  all  these  cases  is  this,  namely,. 
that  the  sides  of  the  triangle  bear  to  each  other  the  same 
lion  as  the  parts  which  they  uepresent. 

And  this  is  called,  Making  every  side  radius. 


OF  HEIGHTS  AND  DISTANCES. 

BY  the  mensuration  and  protraction  of  lines  and  angles,,  are  de- 
termined the  lengths,  heights,  depths,  and  distances  of  bodies  or 
objects. 

Accessible  lines  are  measured  by  applying  to  them  some  certain 
measure  a  number  of  times,  as  an  inch,  or  foot,  or  yard.  But  in- 
accessible lines  must  be  measured  by  taking  angles,  or  by  some 
such  method,  drawn  from  the  principles  of  geometry. 

When  instruments  are  used  for  taking  the  magnitude  of  the 

24 


370 


THE  PRACTICAL  MODEL  CALCULATOR. 


angles  in  degrees,  the  lines  are  then  calculated  by  trigonometry : 
in  the  other  methods,  the  lines  are  calculated  from  the  princi- 
ple of  similar  triangles,  without  regard  to  the  measure  of  the 
angles. 

Angles  of  elevation,  or  of  depression,  are  usually  taken  either 
with  a  theodolite,  or  with  a  quadrant,  divided  into  degrees  and  mi- 
nutes, and  furnished  with  a  plummet  suspended  from  the  centre, 
and  two  sides  fixed  on  one  of  the  radii,  or  else  with  telescopic 
sights. 

To  take  an  angle  of  altitude  and  depression  with  the  quadrant. 

Let  A  be  any  object,  as  the  sun, 
moon,  or  a  star,  or  the'  top  of  a  tower, 
or  hill,  or  other  eminence ;  and  let  it 
be  required  to  find  the  measure  of  the 
angle  ABC,  which  a  line  drawn  from 
the  object  makes  with  the  horizontal 
line  BC. 

Fix  the  centre  of  the  quadrant  in 
the  angular  point,  and  move  it  round 
there  as  a  centre,  till  with  one  eye  at 
D,  the  other  being  shut,  you  perceive  the  object  A  through  the 
sights  :  then  will  the  arc  GH  of  the  quadrant,  cut  off  by  the  plumb 
line  BH,  be  the  measure  of  the  angle  ABC,  as  required. 


The  angle  ABC  of  depression  of  any  ob- 
ject A,  is  taken  in  the  same  manner ;  except 
that  here  the  eye  is  applied  to  the  centre,  and 
the  measure  of  the  angle  is  the  arc  GH,  on 
the  other  side  of  the  plumb  line. 


The  following  examples  are  to  be  constructed  and  calculated  by 
the  foregoing  methods,  treated  of  in  trigonometry.. 

Having  measured  a  distance  of  200  feet,  in  a  direct  horizontal 
line,  from  the  bottom  of  a  steeple,  the  angle  of  elevation  of  its  top, 
taken  at  that  distance,  was  found  to  be  4J°  30':. from  hence  it  is 
required  to  find  the  height  of  the  steeple. 

Construction. — Draw  an  indefinite  line,  upon  which  set  off  AC  = 
200  equal  parts,  for  the  measured  distance.  Erect  the  indefinite 
perpendicular  AB ;  and  draw  CB  so  as  to  make  the  angle  C  » 
47°  30',  the  angle  of  elevation;  and  it  is  done.  Then  AB,  mea- 
sured on  the  scale  of  equal  parts,  is  nearly  218J. 

Calculation. 

As  radius 10-0000000 

To  AC  200 2-3010300 

So  tang,  angle  C  47°  30' 10-0379475       / 

To  AB  218-26  required 2-3389775    <f- 


OF   HEIGHTS   AND   DISTANCES.  371 

• 

What  was  the  perpendicular  height  of  a  cloud,  or  of  a  balloon, 
when  its  angles  of  elevation  were  35°  and  64°,  as  taken  by  two 
observers,  at  the  same  time,  both  on  the  same  side  of  it,  and  in 
the  same  vertical  plane ;  their  distance,  as  under,  being  half  a  mile, 
or  880  yards.  And  what  was  its  distance  from  the  said  two  ob- 
servers ? 

Construction. — Draw  an  indefinite  ground  line,  upon  which  set 
off  the  given  distance  AB  =  880 ;  then  A  and  B  are  the  places 
of  the  observers.  Make  the  angle  A  =  35°,  and  the  angle  B  = 
64° ;  and  the  intersection  of  the  lines  at  C  will  be  the  place  of  the 
balloon ;  from  whence  the  perpendicular  CD,  being  let  fall,  will  be 
its  perpendicular  height.  Then,  by  measurement,  are  found  the 
distances  and  height  nearly,  as  follows,  viz.  AC  1631,  BC  1041, 
DC  936. 

Calculation. 

First,  from     angle     B     64°  "$>'*'*/ 

Take    angle     A     35  /'     / 

Leaves  angle  ACB  29  /*' 


Then,  in  the  triangle  ABC, 

As  sine     angle  ACB    29°       9-6855712 

To  opposite  side  AB     880        2-9444827 

So  sine     angle  A         35° 9-7585913 

To  opposite  side  BC   1041-125 3-0175028 

As  sine     angle  ACB    29°      9-6855712 

To  opposite  side  AB     880        2-9444827 

So  sine     angleB   116°or64° 9-9536602 

To  opposite  side  AC  1631-442  3-2125717 

And,  in  the  triangle  BCD, 

As  sine     angle  D          90°       10-0000000 

To  opposite  side  BC   1041-125 3-0175028 

So  sine      angleB         64°       9-9536602 

•  To  opposite  side  CD     935-757 2-9711630 

Having  to  find  the  height  of  an  obelisk  standing  on  the  top  of  a 
declivity,  I  first  measured  from  its  bottom,  a  distance  of  40  feet, 
and  there  found  the  angle,  formed  by  the  oblique  plane  and  a  line 
imagined  to  go  to  top  of  the  obelisk  41° ;  but,  after  measuring  on 
in  the  same  direction  60  feet  farther,  the  like  angle  was  only  23°  45'. 
What  then  was  the  height  of  the  obelisk  ? 

Construction. — Draw  an  indefinite  line  for  the  sloping  plane  or 
declivity,  in  which  assume  any  point  A  for  the  bottom  of  the 
obelisk,  from  whence  set  off  the  distance  AC  =  40,  and  again 
CD  =  60  equal  parts.  Then  make  the  angle  C  =41°,  and  the 
angle  D  =  23°  45' ;  and  the  point  B,  where  the  two  lines  meet, 
will  be  the  top  of  the  obelisk.  Therefore  AB,  joined,  will  be  its 
height. 


372          THE  PRACTICAL  MODEL  CALCULATOR. 


Calculation. 

From  the  angle  C  41°  00' 
Take  the  angle  D  23  45 
Leaves  the  angle  DBG  17  15 


Then,  in  the  triangle  DBG, 

As  sine      angle  DBG  17°  15' 9-4720856 

To  opposite  side  DC     60         1-7781513 

So  sine      angle  D        24   45 9-6050320 

To  opposite  side  CB     81-488 1-9110977 

And,  in  the  triangle  ABC, 

As  sum  of  sides  CB,  CA  121-488  2-0845333 

To  difference  of  sides     CB,  CA    41-488  1-6179225 

So  tang,  half  sum  angles  A,  B       69°  30'  10-4272623 

To  tang,  half  diff.  angles  A,  B       42    24| 9-9606516 

The  diff.  of  these  is  angle  CBA       27      5J 

Lastly,  as  sine  angle  CBA  27°5J' 9-6582842 

To  opposite  side         CA     40     "  1-6020600 

So  sine  angle  C        41°0'   9-8169429 

To  opposite  side        AB    57*623 1-7607187 

Wanting  to  know  the  distance  between  two  inaccessible  trees,  or 
other  objects,  from  the  top  of  a  tower,  120  feet  high,  which  lay  in 
the  same  right  line  with  the  two  objects,  I  took  the  angles  formed 
by  the  perpendicular  wall  and  lines  conceived  to  be  drawn  from  the 
top  of  the  tower  to  the  bottom  of  each  tree,  and  found  them  to  be 
33°  and  64J°.  What  then  may  be  the  distance  between  the  two 
objects  ? 

Construction. — Draw  the  indefinite 
ground  line  BD,  and  perpendicular  to 
it  BA  =  120  equal  parts.  Then  draw 
the  two  lines  AC,  AD,  making  the  two 
angles  BAG,  BAD,  equal  to  the  given 
angles  33°  and  64}°.  So  shall 
objects. 


BCD 
and  D  be  the  places  of  the  two 


Calculation. — First,  In  the  right-angled  triangle  ABC, 

As  radius 10-0000000 

ToAB , 120  2-0791812 

So  tang,  angle  BAG 33° 9-8125174 

ToBC 77-929  1-8916986 

And,  in  the  right-angled  triangle  ABD, 

As  radius 10-0000000 

ToAB 120    2-0791812 

So  tang,  angle  BAD....  64J° 10-3215039 

To  BD 251-585     2-4006851 

From  which  take  BC    77-929 

Leaves  the  dist.    CD  173-656  as  required. 


SPHERICAL   TRIGONOMETRY. 


373 


Being  on  the  side  of  a  river,  and  wanting  to  know  the  distance 
to  a  house  which  was  seen  on  the  other  side,  I  measured  200  yards 
in  a  straight  line  by  the  side  of  the  river ;  and  then  at  each  end 
of  this  line  of  distance,  took  the  horizontal  angle  formed  between 
the  house  and  the  other  end  of  the  line ;  which  angles  were,  the 
one  of  them  68°  2',  and  the  other  73°  15'.  What  then  were  the 
distances  from  each  end  to  the  house  ? 

Construction. — Draw  the  line  AB  =  200  equal  parts.     Then 


draw  AC  so  as  to  make 
the  angle  B  =  73°  15'. 
house  required. 


To 


the  angle  A  =  68°  2',  and  BC  to  make 
So  shall  the  point  C  be  the  place  of  the 


Calculation. 


68° 
73 
141 

180 


the  given  angle  A 
Add  the  given  angle  B 
Then  their  sum 
Being  taken  from 
Leaves  the  third  angle  C    38 
Hence,  As  sin.  angle  C  38 
To  op.  side  AB  ~ 
sin. 
op. 


2' 
15 
17 

0 
43 


And, 


0  43' 9-7962062 

200       2-3010300 

So  sin.  angle  A  68°   2' 9-9672679 

To  op.  side   BC  296-54 2-4720917 

As  sin.  angle  C  38°  43' 9-7962062 

To  op.  side  AB  200       2-3010300 

So  sin.  angle  B  73°  15' 9-9811711 

To  op.  side  AC  306-19 2-4859949 


SPHERICAL  TRIGONOMETRY. 

This  Article  is  taken  from  a  short  Practical  Treatise  on  Spherical  Trigonometry, 
by  Oliver  Byrne,  the  author  of  the  present  work.  Published  by  J.  A.  Valpy. 
London,  1835. 

As  the  sides  and  angles  of  spherical  triangles  are  measured  by 
circular  arcs,  and  as  these  arcs  are  often  greater  than  90°,  it  may 
be  necessary  to  mention  one  or  two  particulars  respecting  them. 

The  arc  CB,  which  when  added  to 
AB  makes  up  a  quadrant  or  90°,  is 
called  the  complement  of  the  arc  AB ; 
every  arc  will  have  a  complement, 
even  those  which  are  themselves 
greater  than  90°,  provided  we  con- 
sider the  arcs  measured  in  the  direc- 
tion ABCD,  &c.,  as  positive,  and 
consequently  those  measured  in  the 
opposite  direction  as  negative.  The 
complement  BC  of  the  arc  AB  com- 
mences at  B,  where  AB  terminates, 
and  may  be  considered  as  generated  by  the  motion  of  B,  the  ex- 
2  G 


374  THE   PRACTICAL  MODEL   CALCULATOR. 

tremity  of  the  radius  OB,  in  the  direction  BC.  But  the  comple- 
ment of  the  arc  AD  or  DC,  commencing  in  like  manner  at  the  ex- 
tremity D,  must  be  generated  by  the  motion  of  D  in  the  opposite 
direction,  and  the  angular  magnitude  AOD  will  here  be  diminished 
by  the  motion  of  OD,  in  generating  the  complement ;  therefore 
the  complement  of  AOD  or  of  AD  may  with  propriety  be  consi- 
dered negative. 

Calling  the  arc  AB  or  AD,  0,  the  complement  will  be  90°  —  e ; 
the  complement  of  36°  44'  38"  is  53°  15'  27" ;  and  the  complement 
of  136°  27'  39"  is  negative  46°  27'  39". 

The  arc  BE,  which  must  be  added  to  AB  to  make  up  a  semi- 
circle or  180°,  is  called  the  supplement  of  the  arc  AB.  If  the  arc 
is  greater  than  180°,  as  the  arc  ADF  its  supplement,  FE  mea- 
sured in  the  reverse  direction  is  negative.  The  expression  for  the 
supplement  of  any  arc  o  is  therefore  180°  —  e ;  thus  the  supple- 
ment of  112°  29'  35"  is  67°  30'  25",  and  the  supplement  of  205° 
42'  is  negative  25°  42'. 

In  the  same  manner  as  the  complementary  and  supplementary 
arcs  are  considered  as  positive  or  negative,  according  to  the  di- 
rection in  which  they  are  measured,  so  are  the  arcs  themselves 
positive  or  negative ;  thus,  still  taking  A  for  the  commencement, 
or  origin,  of  the  arcs,  as  AB  is  positive,  AH  will  be  negative.  In 
the  doctrine  of  triangles,  we  consider  only  positive  angles  or  arcs, 
and  the  magnitudes  of  these  are  comprised  between  9  =  0  and  o  = 
180°  ;  but  in  the  general  theory  of  angular  quantity,  we  consider 
both  positive  and  negative  angles,  according  as  they  are  situated 
above  or  below  the  fixed  line  AO,  from  which  they  are  measured, 
that  is,  according  as  the  arcs  by  which  they  are  estimated  are  posi- 
tive or  negative.  Thus  the  angle  BOA  is  positive,  and  the  angle 
AOH  negative.  Moreover,  in  this  more  extended  theory  of  angular 
magnitude,  an  angle  may  consist  of  any  number  of  degrees  what- 
ever ;  thus,  if  the  revolving  line  OB  set  out  from  the  fixed  line  OA, 
and  make  n  revolutions  and  a  part,  the  angular  magnitude  gene- 
rated is  measured  by  n  times  360°,  plus  the  degrees  in  the  ad- 
ditional part. 

In  a  right-angled  spherical  triangle  we  are  to  recognise  but  five 


parts,  namely,  the  three  sides  #,  5,  <?,  and  the  two  angles  A,  B ; 
BO  that  the  right  angle  C  is  omitted. 


SPHERICAL   TRIGONOMETRY. 


375 


Let  A',  cr,  B,'  be  the  comple- 
ments of  A,  c,  B,  respectively, 
and  suppose  b,  a,  B',  cf,  A',  to  be 
placed  on  the  hand,  as  in  the 
annexed  figure,  and  that  the 
fingers  stand  in  a  circular  order, 
the  parts  represented  by  the 
fingers  thus  placed  are  called 
circular  parts. 

If  we  take  any  one  of  these  as 
a  middle  part,  the  two  which  lie 
next  to  it,  one  on  each  side,  will 
be  adjacent  parts.  The  two  parts 
immediately  beyond  the  adjacent 
parts,  one  on  each  side,  are  called 
the  opposite  parts. 

Thus,  taking  A'  for  a  middle  part,  b  and  cr  will  be  adjacent  parts, 
and  a  and  B'  are  opposite  parts. 

If  we  take  c'  as  a  middle  part,  A'  and  B'  are  adjacent  parts,  and 
b,  a,  opposite  parts. 

When  B'  is  a  middle  part,  <?',  a,  become  adjacent  parts,  and  A', 
b,  opposite  parts. 

Again,  if  we  take  a  as  a  middle  part,  then  B',  b,  will  be  adjacent 
parts,  and  cf,  A',  opposite  parts. 

Lastly,  taking  b  as  a  middle  part,  A',  a,  are  adjacent  parts,  and 
c',  B',  opposite  parts. 

This  being  understood,  Napier's  two  rules  may  be  expressed  as 
follows : — 

I.  Rad.  X  sin.  middle  part  =  product  of  tan.  adjacent  parts. 

II.  Rad.  X  sin.  middle  part  =  product  of  cos.  opposite  parts. 
Both  these  rules  may  be  comprehended  in  a  single  expression,  thus, 

Rad.  sin.  mid.  =  prod.  tan.  adja.  =  prod.  cos.  opp. ; 
and  to  retain  this  in  the  memory  we  have  only  to  remember,  that 
the  vowels  in  the  contractions  sin.,  tan.,  cos.,  are  the  same  as  those 
in  the  contractions  mid.,  adja.,  opp.,  to  which  they  are  joined. 

These  rules  comprehend  all  the  succeeding  equations,  reading 
from  the  centre,  R  =  radius. 

In  the  solution  of  right-angled  spherical  triangles,  two  parts  are 
given  to  find  a  third,  therefore  it  is  necessary,  in  the  application  of 
this  formula,  to  choose  for  the  middle  part  that  which  causes  the 
other  two  to  become  either  adjacent  parts  or  opposite  parts. 
In  a  right-angled  spherical  triangle,  the  hypothenuse 
c   —  61°    4'  56" ;  and  the  angle 
A  —  61°  50'  29".     Required  the  adjacent  leg? 


A. 


90° 
61 

0' 
4 

00 
56 

n 

A 

90° 
-61 

0' 
50 

00" 

29 

28 

55 

04 

=  c'. 

28 

y 

31  = 

376 


THE    PRACTICAL   MODEL    CALCULATOR. 


V — Tan.  o.( 


I 


In  this  example,  A^  is  selected  for  the  middle  part,  because  then 
b  and  c'  become  adjacent  parts,  as  in  the  annexed  figure. 

Rad  x  sin.  A'  =  tan.  b  X  tan.  <?'. 
rad.  X  sin.  A' 


tan.  b  = 


tan  <?' 


jffy  Logarithms. 

Rad.   - -10-0000000 

Sin.  A/-28°9/21//-  9-6738628 

19-6738628 
Tan.  c/-28°55/4^-  9-7422808 

Tan.6/~40°30/16//-9-9315820 
The  side  adjacent  to  the  given 
angle  is  acute  or  obtuse,  accord- 
ing as  the  hypothenuse  is  of  the 
same,  or  of  different  species  with  the  given  angle. 

.-.  the  leg  b  =  40°  30r  16",  acute. 

Supposing  the  hypothenuse  c  =  113°  55',  and  the  angle  A  =  31°  51', 
then  the  adjacent  leg  b  would  be  117°  34',  obtuse. 


SPHERICAL   TRIGONOMETRY. 


377 


In  the  right-angled  spherical  triangle  ABC,  are  given  the  hypo- 
thenuse  e  =  113°  55',  and  the  angle  A  =  104°  08';  to  find  the 
opposite  leg  a. 

c  =  113°  55' 

90      0 


23    55 

A  =  104°  08' 
90      0 


14    08  =  A'. 


In  this  example,  a  is  taken  for  the  middle  part,  then  A'  and  cf 
are  opposite  parts.     (See  the  subjoined  figure/ 

From  the  general  formula,  we 
have, 

Had.  X  sin.  a  =  cos.  A'  X  cos.  ef. 
cos.  A'  x  cos.  c' 


sin.  a  = 


Rad. 


By  Logarithms. 

cos.  A'  -  14°  08' 9-9860509 

cos.  c'   -23    55 9-9610108 

19-9476617 
Radius 10-0000000 


sm 


•9476617 


f  117°  34' 
m-  d\    62    26 

The  obtuse  side  117°  34'  is  the  leg  required,  for  the  side  oppo- 
site to  the  given  angle  is  always  of  the  same  species  with  the 
given  angle. 

If  in  a  right-angled  spherical 
triangle,  the  hypothenuse  were 
78°  20',  and  the  angle  A  = 
37°  25',  then  the  opposite  leg 
a  =  36°  31',  and  not  143°  29', 
because  the  given  angle  is  acute. 

In  a  right-angled  spherical  tri- 
angle, are  given  c  =  78°  20',  and 
A  =  37°  25',  to  find  the  angle  B. 

90°    0' 
c  —  78    20 
TT~!o  =  c'. 

90°    0' 
A  =  37    25 

52    35=  A' 
2  G  2 


378 


THE  PRACTICAL  MODEL  CALCULATOR. 


Here  the  complement  of  the  hypothenuse*(c')  is  the  middle  part ; 
and  the  complement  of  the 
angle  opposite  the  perpen- 
dicular (A'),  and  the  com- 
plement of  the  angle  oppo- 
site the  base  (B')  are  the 
adjacent  parts.  This  will 
readily  be  perceived  by 
reference  to  the  usual 
figure  in  the  margin. 

Rad.  X  sin.  cf  =  tan.  A' 
X  tan.  B'; 

Rad.  X  sin.  cf 

.•.tan.B'  =  -      — r; — . 
tan.  A' 

By  Logarithms. 

Rad 10-0000000 

sin.  c'  -  11°  40*.  9-3058189 
19-3058189 

tan.  A'  -  52°  35'  10-1163279 
.-.  tan.  B'  -  8°  48'  9-1894910 

But  90  -  B   =  B' 
hence  90  —  B'  =  B. 
90°    0' 
8    48 
B  =  81°  12'. 

When  the  hypothenuse  and  an  angle  are  given,  the  other  angle  is 
acute  or  obtuse,  according  as  the  given  parts  are  of  the  same  or  of 
different  species. 

In  the  above  example,  both  the  given  parts  are  acute,  therefore 
the  required  angle  is  acute  ;  but  if  one  be  acute  and  the  other  ob- 
tuse, then  the  angle  found  would  be  obtuse  : — Thus,  if  the  hypo- 
thenuse be  113°  55',  and  the  angle  A  =  31°  51' ;  then  will  B'  = 
14°  08',  and  the  angle  B  =  104°  08'. 

Given  the  hypothenuse  c  =  61°  04'  56",  and  the  side  or  leg, 
a  =  40°  30'  20",  to  find  the  angle  adjacent  to  a.  </ 

90°    0'    0" 
c  =  61    04  56 

28    55  04"  =  c". 

The  three  parts  are  here 
connected ;  therefore  the  com- 
plement of  B  is  the  middle 
part,  a  and  the  complement  of 
c  are  the  adjacent  parts. 

Hence  we  have, 
Rad.  X  sin.  B'  =  tan.  a  X  tan.  c'. 

tan.  a  X  tan.  <?' 
.-.  sin.  B'  =  — 


SPHERICAL   TRIGONOMETRY. 


379 


By  Logarithms. 


tan.  a  —  40°  30'  20" 
tan.  e'  -  28    55  04 


9-9315841 
9-7422801 


19-6738642 
Had 10-OOQQQOQ 

sin.  B'....28°  09'  31" 9-6738642 

90°    0'    0" 
B'  =  28    09  31 

61    50  29  =  B. 

The  angle  adjacent  to  the  given  side  is  acute  or  obtuse  accord- 
ing as  the  hypothenuse  is  of  the  same  or  of  different  species  with 
the  given  side. 

Before  working  the  above  example,  it  was  easy  to  foresee  that 
the  angle  B  would  be  acute ;  but  suppose  the  hypothenuse  =  70° 
20',  and  the  side  a  =  117°  34',  then  the  angle  B  would  be  obtuse, 
because  a  and  c  are  of  different  species. 

RULE  V. — In  a  spherical  triangle,  right-angled  at  (?,  are  given 
c  =  78°  20'  and  b  =  117°  34',  to  find  the  angle  B ;  opposite  the 
given  leg,  (see  the  next  diagram.) 

In  this  example,  b  becomes  the  middle  part,  and  cf  and  B'  oppo- 
site parts ;  and  therefore,  by  the  rule, 

Had.  x  sin.  b  =  cos.  B'  x  cos.  c' ;  that  is, 

Rad.  X  sin.  b 
cos.  B'  = 


cos.  c' 
90°  -  78°  20'  =  11°  40' 


c'. 


JTence,  ly  Logarithms. 

Rad 10-0000000 

sin.  b  =  sin.  117°  34'  \ 

or  sin.    62    26  / 


9-9476655 


19-9476655 
cos.  c'  11°  40' 9-9909338 

cos.  B'  25°  09'..,  '.  9-9567317 


380  THE    PRACTICAL   MODEL   CALCULATOR. 

B 

But  since  the  angle 
opposite  the  given 
side  is  of  the  same 
species  with  the  given 
side,  90°  must  be 
added  to  B',  to  pro-  a^ 
duce  B  :—  viz.  90°  -f 
25°  09'  =  115°  09'. 

Given  e  =  61°  04' 
56",  and  b  =  40°  30' 
20",  to  find  the  other 
side  a. 

Here  ef  is  the  mid- 
dle part,  a  and  b  the 
opposite  parts  ;  hence 
by  position  4,  a  =  50°  30'  30". 

Given  the  side  b  =  48°  24'  16",  and  the  adjacent  angle  A  = 
66°  20'  40",  to  find  the  side  a. 

In  this  instance,  b  is  the  middle  part,  the  complement  of  A  and 
a  are  adjacent  parts.  Consequently,  a  =  59°  38'  27". 


In  the  right-angled  spherical  triangle  ABC, 

S  T^  32'  55"  }*>  **  *» 

Answer,  66°  20'  40". 


A. 


The  required  angle  is  of  the  same  species  as  the  given  side,  and 
vice  versa. 

Given  the  side  b  =  49°  17',  and  its  adjacent  angle  A  =  23°  28', 
to  find  the  hypothenuse. 

Making  A'  the  middle  part,  the  others  will  be  adjacent  parts, 
and,  therefore,  by  the  first  rule  we  have  c  =  51°  42'  37". 

In  a  spherical  triangle,  right-angled  at  C,  are  given.  b  =  29°  12' 
50",  and  B  =  37°  26'  21",  to  find  the  side  a. 

Taking  a  for  the  middle  part,  the  other  two  will  be  adjacent  parts  ; 
hence  by  the  rule, 

Rad.  X  sin.  a  =  tan.  b  X  tan.  B' 
that  is,  rad.  X  sin.  a  —  tan.  b  x  cot.  B 

tan.  b  X  cot.  B 
.*.  sin.  a  =  -  T  -- 
rad. 

In  this  case,  there  are  two  solutions,  i.  e.  a  and  the  supple- 
ment of  a,  because  both  of  them  have  the  same  sine.  As  sin.  a 
is  necessarily  positive,  b  and  B  must  necessarily  be  always  of 
the  same  species,  so  that,  as  observed  before,  the  sides  including 
the  right  angle  are  always  of  the  same  species  as  the  opposite 
angles. 


SPHERICAL  TRIGONOMETRY. 


381 


In  working  this  example, 
we  find  the  log.  sin.  a  = 
9-8635411,    which    corre- 
sponds to  46°  55'  02", 
or,  133°  04'  58". 

It  appears,  therefore, 
that  a  is  ambiguous,  for 
there  exist  two  right-angled 
triangles,  having  an  oblique 
angle,  and  the  opposite  side 
in  the  one  equal  to  an 
oblique  angle  and  an  oppo- 
site side  in  the  other,  but 
the  remaining  oblique  angle 
in  the  one  the  supplement 
of  the  remaining  oblique 
angle  in  the  other.  These  triangles  are  situated  with  respect 
to  each  other,  on  the  sphere,  as  the  triangles  ABC,  ADC, 
in  the  annexed  diagram,  in  which,  with  the  exception  of  the 
common  side  AC,  and  the  equal  angles  B,  D,  the  parts  of  the 
one  triangle  are  supplements  of  the  corresponding  parts  of  the 
other, 

In  a  right-angled  spherical  triangle  are 

/the  side  a  .......  ......  =  42°  12',  \to  find  the  adjacent 

\  its  opposite  angle  A  =  48°         J  angle  B. 

The  complement  of  the  given  angle  is  the  middle  part;  and 
neither  a  nor  B'  being  joined  to  A',  they  are  consequently  opposite 
parts  ;  hence,  the  angle  B  =  64°  35',  or  115°  25'  ;  this  case,  like 
the  last,  being  ambiguous,  or  doubtful. 

Given  a  =  11°  30',  and  A  =  23°  30',  to  find  the  hypothenuse  c. 
c  =  30°,  or  150°,  being  ambiguous. 

In  a  right-angled  triangle,  there  are  given  the  two  perpendicu- 
lar sides,  viz.  a  =  48°  24'  16",  b  =  59°  38'  27",  to  find  the 
angle  A. 

A  =  66°  20'  40". 

Given  a  =  142°  31',  b  =  54°  22',  to  find  c. 
c  =  117°  33'. 


11 


Given  < 


Given 


81  ¥>    \  -^e(lu^re^  tne  s^e  a- 
a  =  36°  31'. 

66°  20'  40"  1 

52    39   55    V  to  find  the  hypothenuse  c. 

c  =  70°  23'  42". 


382 


THE  PRACTICAL  MODEL  CALCULATOR. 


MEASTTKEMENT  OF  ANGLES. 

From  the  "  Civil  Engineer  and  Architect's  Journal "  for  Oct.  and  Nov.  1847. 

A  NEW  METHOD  OP  MEASURING  THE  DEGREES,  MINUTES,  ETC.,  IN  ANY 
RECTILINEAR  ANGLE  BY  COMPASSES  ONLY,  WITHOUT  USING  SCALE  OR 
PROTRACTOR. 

APPLY  AB  =  #,  from  B  to  1 ;  from  1  to  2 ;  from  2  to  3 ;  from 
3  to  4 ;  from  4  to  5.  Then  take  B  5,  in  the  compasses,  and  apply 
it  from  B  to  6  ;  from  6  to  7  ;  from  7  to  8 ;  from  8  to  9  ;  and  from 
9  to  10,  near  the  middle  of  the  arc  AB.  With  the  same  opening, 


B  5  or  A  4,  or  ?/,  which  we  shall  term  it,  lay  off  4,11,  11,12,  and 
12,13.  Then  the  arc  between  13  and  10  is  found  to  be  contained 
23  times  in  the  arc  AB. 


MEASUREMENT   OF   ANGLES. 


383 


Hence,  we  have,      5  x  —  y  =  360° ; 

9#  -f  z  =  x\  x 

23  z  =  2; ;  or,  .z  =  ^3. 

#  22  # 

— •>  ^u  ( 

By  substituting  this  value  in  the  first  equation,  we  obtain, 

22  x 
ox  —  ^7^  —  360. 


Apply  AB  =  x,  from  B  to  1  ;  from  1  to  2  ;  from  2  to  3  ;  from 
3  to  4.  Then  take  B  4,  in  the  compasses,  and  apply  it  on  the  arc, 
from  B  to  4  ;  from  4  to  5  ;  from  5  to  6  ;  from  6  to  7  ;  and  from 
7  to  8,  near  the  middle  of  the  arc  AB.  With  the  same  opening, 
B  4  =  y,  lay  off  A  9,  9,10,  10,11,  11,12,  12,13,  and  13,14.  The 
arc  between  14  and  8  is  found  to  be  contained  nearly  24  times  in 
the  arc  AB.  Therefore,  we  have, 

4  x  +  y  =  360; 


242 
x 


x\  or,  z  =  24. 


Substituting  this  value  of  y  in  the  first  equation, 

25*       360; 


264 
360  x  264 

1071 


88°  44'-333. 


How  to  lay  off  an  angle  of  any  number  of  degrees,  minutes, 
with  compasses  only,  without  the  use  of  scale  or  protractor. 

Let  it  be  required 

to  lay  off  an  angle  of  ^  B 

36°  40'  =  ]8.  Take  any 
small  opening  of  the 
compasses  less  than 
one-tenth  of  the  ra- 
dius, and  lay  off  any 
number  of  equal  small 
arcs,  from  A  to  1 ; 
from  1  to  2  ;  from  2  to 
3,  &c.,  until  we  have 
laid  off  an  arc,  AB, 
greater  than  the  one 
required.  Draw  B  b 
through  the  centre  o, 
tli en  will  the  arc  a  b  = 
arc  AB,  which  we  shall 


384 


THE  PRACTICAL  MODEL  CALCULATOR. 


put  =  20  $  in  this  example,  and  proceed  to  measure  a  b  as  in  the 
first  example.  Lay  off  a  b  from  b  to  c  ;  from  c  to  d  ;  from  d  to  e  ; 
from  e  to/;  from  /to  g.  Putting  g  a  —  A1?  then, 

108 
6  X  20*  +  At  =  360°  =  --p;  because, 


360°         21600 


108 
TP 


36°  40'  ""  2200 
Lay  off,  as  before  directed,  g  a,  =  Al5  from  a  to  A,  from  h  to  s, 
and  b  to  t ;  then  calling  s  £,  Aa?  we  have 

and  we  find  that  s  t  is  contained  28  times  in  the  arc  a  b ; 

-J  AO 

.-.  120*+  Aa  =  -jj- P;  3  Aa  +  Aa  =  20*;  and  28  A3  =  20*. 

Eliminating  Aa  and  A  &  we  find 

29205 
3  =   2268  ft  =  12*9  times  *  nearly ; 

.-.  36°  40'  =  L  A  o  N  is  laid  off  with  as  much  ease  and  certainty 
as  by  a  protractor. 

As  a  second  example,  let  it  be  required  to  lay  off  an  angle  of 
132°  27'.     From  180°  0'  take  132°  27'  =  47°  33',  which  put  =  ^ 

360°         2400  v  v 

^HTQ  QO/  =  -O-IIT-  when  put  =  ^  then  gi3  =  360°  =  ft. 

29 


We  have  laid  off  29  small  arcs  from  A  to  B  ;  29  =  «.     AB  = 
b  =  bc  =  cd  =  de='cf.  And  ag  =  bh  =  af  =  &±;  Jig  —  A3- 

.-.  5  x  29*  +  At  =  360°  =  jP  =  we  »  db  A!        (1) 
2  A4  -  A3  =  29 1,  or  n  A,  ±  A2  =  «*          (2) 
or  =  ^ft  3 


MEASUREMENT    OF   ANGLES.  385 

Eliminating  Ax  and  An,  we  have 

{0tttydr(g=Fl)}«8          {5-2-13  +  (13  +  1)}29-317 
*  ~~  vnq  2400-2-13 

1323729 

62400    *  =  ^^i"  ^mes  *  very  nearty'     Hence  the  line  0  N  deter- 
mines the  angle  a  o  N  =  132°  27'. 
In  the  expression 


substituting  the  numerals  of  the  first  example,  then 

{6-3-28  +  (28  -  1)}20-11         29205 

*  "  ~  108-3-28  -  *  =  =  "2268"  *  ™  12*9  times  *  nearl?> 

the  result  before  obtained. 

The  ambiguous  signs  of  (R)  cannot  be  mistaken  or  lead  to  error, 
if  the  manner  in  which  it  is  deduced  from  (1),  (2),  (3),  be  attended 
to.  From  (3) 

C  tp 

A3  =  —  J  substituting  this  value  of  A3»  in  (2),      » 

wA1=ft=FA3  =  ftrF  — >  which,  when  substituted  for  A! 
in  (1),  gives 

V  1     /  £  3>\ 

- - /3  =  m  s<t>  ±  -  (^E  <j>  qn  — J  ;  from  which  (R)  is  found. 

This  method  of  measuring  angles  is  more  exact  than  it  may  ap- 
pear ;  for  if,  in  the  first  example,  we  take 

5  x  —  y  =  360;  9#  +  z  =  a;  ;•  and  20  z  =  x, 
then  a;  =  4   i~  =  73°  33'  85. 


The  first  equations  gave  73°  33'  82  when  23  z  =  x,  so  it  does 
not  matter  much  whether  20,  21,  22,  23,  24,  or  25  times  z  make 
x.  This  fact  is  particularly  worth  attention. 

Given  the  three  angles  to  find  the  three  sides. 

The  following  formulas  give  any  side  a  of  any  spherical  triangle. 

—  cos.  I  S  cos.  (}  S  -  A) 
sm.|a  =  v/-       —  sin7B-sin.^-      ~'  and 
cos.  (J  S  -  B)  cos.  (J  S  —  C) 
qos.  I  a  ==  v/-  Sin.  B  sin.  C. 

Given  the  three  sides  to  find  the  three  angles. 

,-„  i  A       >•  ft  s  -  J)  sin-  (i  s  -  *) 
Bm.  j  A  =  v/-     " 


sin.  |  S  sin.  (J  S  —  a] 

cos.  *  A  =  v/  -  •  —  i  —  •  - 
sm.  b  sm.  c. 

23 


386 


GRAVITY—  WEIGHT—  MASS. 

SPECIFIC  GRAVITY,  CENTRE  OP  GRAVITY,  AND  OTHER  CENTRES  OF  BODIES. 
-  WEIGHTS  OF  ENGINEERING  AND  MECHANICAL  MATERIALS.  -  BRASS, 
COPPER,  STEEL,  IRON,  WATER,  STONE,  LEAD,  TIN,  ROUND,  SQUARE,  FLAT, 
ANGULAR,  ETC. 

1.  IN  a  second,  the  acceleration  of  a  body  falling  freely  in  vacuo 
is  32*2  feet;  what  velocity  has  it  acquired  at  the  end  of  5  seconds  ? 

32-2  x  5  =  161  feet,  the  velocity. 

2.  A  cylinder  rolling  down  an  inclined  plane  with  an  initial  velo- 
city of  24  feet  a  second,  and  suppose  it  to  acquire  each  second  5  ad- 
ditional feet  velocity  ;  what  is  its  velocity  at  the  end  of  3  -7  seconds  ? 

24  +  3-7  x  5  =  42-5  feet. 

3.  Suppose  a  locomotive,  moving  at  the  rate  of  30  feet  a  second, 
(as  it  is  usually  termed,  with  a  30  feet  velocity,)  and  suppose  it  to  lose 
5  feet  velocity  every  second  ;  what  is  its  velocity  at  the  end  of  3*33 
second^  ? 

The  acceleration  is  —  3-33,  negative. 

.-.  30  -  5  x  3-33  =  13-35  feet. 

4.  If  a  body  has  acquired  a  velocity  of  36  feet  in  11  seconds, 
by  uniformly  accelerated  motion  ;  what  is  the  space  described  ? 

36  x  11 
—  g  -  =  198  feet. 

5.  A  carriage  at  rest  moves  with  an  accelerated  motion  over  a 
space  of  200  feet  in  45  seconds  ;  at  what  velocity  does  it  proceed 
at  the  beginning  of  the  46th  second  ? 

200  x  2 

45       ==  8-8889  feet,  the  velocity  at  the  end  of  the  45th  second. 

The  four  fundamental  formulas  of  uniformly  accelerated  motion  are 
v  t  pt2  v2 

v=pt;    *=2-;    «  =  -2-;     s  =  %^' 

v  the  velocity,  p  the  acceleration,  t  the  time,  and  s  the  space. 

6.  What  space  will  a  body  describe  that  moves  with  an  accele- 
ration of  11'5  feet  for  10  seconds. 

fee, 


7.  A  body  commences  to  move  with  an  acceleration  of  5*5  feet, 
and  moves  on  until  it  is  moving  at  the  rate  of  100  feet  a  second  ; 
what  space  has  it  described  ? 


GRAVITY  —  WEIGHT  —  MASS.  387 

8.  A  body  is  propelled  with  an  initial  velocity  of  3  feet,  and  with 
an  acceleration  of  8  feet  a  second;  what  space  is  described  in 
13  seconds  ? 


3  x  13  4-  8  X2V"X   =  715  feet. 

9.  What  distance  will  a  body  perform  in  35  seconds,  commenc- 
ing with  a  velocity  of  10  feet,  and  being  accelerated  to  move  with 
a  velocity  of  40  feet  at  the  beginning  of  the  36th  second  ? 

10  +  40 
— x  35  =  875  feet,  the  distance. 

The  formulas  for  a  uniformly  accelerated  motion,  commencing 
with  a  velocity  £,  are  as  follow : — 

p  t2  c  +  v  v2  —  c2 

2  2  2  p 

The  succeeding  formulas  are  applicable  for  a  uniformly  retarded 
motion  with  an  initial  velocity  o. 

c  +  v  c2  —  v2 


=     —,  —  -^-  —  2~  g       . 

10.  A  body  rolls  up  an  inclined  plane,  with  an  initial  velocity 
of  50  feet,  and  suffers  a  retardation  of  10  feet  the  second  ;  to  what 
height  will  it  ascend  ? 

50 

—  =  5  seconds,  the  time. 

(50)2 
o       -.Q  =  125  feet,  the  height  required. 

The  free  vertical  descent  of  bodies  in  vacuo  offers  an  important 
example  of  uniformly  accelerated  motion.  The  acceleration  in  the 
previous  examples  was  designated  by  p,  but  in  the  particular  mo- 
tion, brought  about  by  the  force  of  gravity,  the  acceleration  is 
designated  by  the  letter  g,  and  has  the  mean  value  of  32-2  feet. 

If  this  value  of  g  be  substituted  for  p,  in  the  preceding  formula, 
we  have, 

t>==32-2  x£;  v  =  8-024964  xv's;  a  =  16-1  x£2;  s  =  -015528  xv2; 
*  =  -031056  x  v  ;  and  *  =  -2492224  x  Vs. 

11.  What  velocity  will  a  body  acquire  at  the  end  of  5  seconds, 
in'  its  free  descent  ? 

32-2  x  5  =  161  feet. 

12.  What  velocity  will  a  body  acquire,   after  a  free  descent 
through  a  space  of  400  feet  ? 

8-024964  x  v/400  =  160-49928  feet. 

13.  What  space  will  a  body  pass  over  in  its  free  descent  during 
10  seconds  ? 

.     16-1  x  (10)2  =  1610  feet. 


388  THE   PRACTICAL   MODEL   CALCULATOR. 

14.  A  body  falling   freely  in  vacuo,  has   in  its  free    descent 
acquired  a  velocity  of  112  feet  ;  what  space  is  passed  over  ? 

•015528  x  (112)2  =  194-783232  feet. 

15.  In  what  time  will  a  body  falling  freely  acquire  the  velocity 
of  30  feet? 

•031056  X  30  =  -93168  seconds. 

16.  In  what  time  will  a  body  pass  over  a  space  of  16  feet,  fall- 
ing freely  in  vacuo  ? 

•2492224  x  ^16  =  -9968896  seconds. 

If  the  free  descent  of  bodies  go  on,  with  an  initial  velocity, 
which  we  may  call  c,  the  formulas  are, 

c  +  32-2x£;  v  =  ^/c2  +  2gs;  v  =  v/c24-  64-4  Xs; 

V2  _  02 

=  ct  +  16'lxt2;  8  =  -%  -  =  -015528  (v2  -  c2). 

If  a  body  be  projected  vertically  to  height,  with  a  velocity  which 
we  shall  term  <?,  then  the  formulas  become, 

t2 
v  =  c  —  32-2  x  t  ;  v  =  */c2  —  64-4  x  s  ;  s  =  ct  —  g  %  = 

ct  -  16-1  x  t2',  s  =  ^^  =  -015528  (c2  -  v2). 

17.  What  space  is  described  by  a  body  passing  from  18  feet  velo- 
city to  30  feet  velocity  during  its  free  descent  in  vacuo. 

From  the  annexed  table,  we  find  that  the  height  due  to  30  feet 
velocity  ....................................................  =  13-97516 

The  height  due  to  18  ................................  =  __^03106 

Space  described  .......................................        8-94410 

Since  this  problem  and  table  are  often  required  in  practical  me- 
chanics, we  shall  enter  into  more  particulars  respecting  it. 


if  we  put  h  =  height   due   to   the   initial   velocity  c\   that   is, 

h  =  H—  ;  and  h  =  the  height  due  to  the  terminal  velocity  v  ;  that  is, 
*9 

^  =  |^;  then, 

s  =  h±  —  A,  for  falling  bodies,  as  in  the  last  example  ;  and 

s  =  h  —  ht,  for  ascending  bodies. 

Although  these  formulas  are  only  strictly  true  for  a  free  descent 
in  vacuo,  they  may  be  used  in  air,  when  the  velocity  is  not  great. 
The  table  will  be  found  useful  in  hydraulics,  and  for  other  heights 
and  velocities  besides  those  set  down,  for  by  inspection  it  is  seen 
that  the  height  -201242  answers  to  the  velocity  3-6  ;  and  the  height 
20-12423  to  36  ;  and  the  height  2012423  to  360  ;  and  so  on. 


WEIGHT — GRAVITY — MASS. 


389 


TABLE  of  the  Heights  corresponding  to  different  Velocities,  in  feet 

the  second. 


-Velocity 
in  Feet. 

CORRESPONDING  HEIGHT  IN  FEET. 

0 

1 

2 

3 

4 

5 

6 

7 

8  |  9 

0 

•000000 

•000155 

•000621 

•001398 

•002484 

•003882 

•005590 

•007609 

•009938 

•012578 

1 

•015528 

•018789 

•020652 

•026242 

•0304348 

•0349379 

•039752 

•044876 

•050311 

•056056 

2 

•062112 

•068478 

•075155 

•082143 

•089441 

•097050 

•104969 

•113199 

•121739 

•130590 

I 

•139752 

•149224 

•159006 

•169099 

•187888 

•190217 

•201242 

•212577 

•224224 

•236180 

4 

•248447 

•261025 

•273913 

•285714 

•300621 

•314441 

•328572 

•343013 

•357764 

•372826 

5 

•388199 

•403882 

•419877 

•436180 

•452795 

•469720 

•486956 

•504503 

•522360!  -550578 

0 

•559006 

•577795 

•596894 

•616304;  -636025 

•656060 

•676397 

•697050 

•718013  -739286 

7 

•760870 

•782764 

•804970 

•827484 

•850310 

•873447 

•896895 

•920652 

•944721 

•969099 

8 

•993789 

1-018790 

1-044100 

1-069720  i  1-095652 

1-121895  ,1-148421 

1-175311 

1-201482 

1-229971 

9 

1-257764 

1-285869 

1-314285 

1-343012 

1-372050 

1-401400  1-431055 

1-461025 

1-491304 

1-521894 

The  following  extension  is  obtained  from  the  foregoing  table, 
by  mere  inspection,  and  moving  the  decimal  point  as  before  di- 
rected. 


>>*i 

>>.» 

K*; 

>>« 

11 

Corresponding 
Height  in  Feet. 

11 

Corresponding 
Height  iu  Feet. 

11 

Corresponding 
Height  in  Feet. 

11 

Corresponding 
Height  in  Feet. 

£.2 

>£ 

£.2 

>.S 

10 

1-552795 

19 

5-60559 

28 

12-17392 

37 

21-25777 

11 

1-878882 

20 

6-21118 

29 

13-05901 

38 

22-42236 

12 

2-065218 

21 

6-84783 

30 

13-97516 

39 

23-61802 

13 

2-624224 

22 

7-51553 

31 

14-92237 

40 

24-84472 

14 

3-043478 

23 

8-21429 

32 

15-90062 

41 

26-10249 

15 

3-49379 

24 

8-94410 

33 

16-90994 

42 

27-39131 

16 

3-97516 

25 

9-70497 

34 

18-78883 

43 

28-57143 

17 

4-48758 

26 

10-49690 

35 

19-02174 

44 

30-06212 

18 

5-03106 

27 

11-31988 

36 

20-12423 

45 

31-4441 

18.  What  mass  does  a  body  weighing  30268  Ibs.  contain  ? 

30268       302680 

=  940  Ibs. 


32-2 


322 


For  the  mass  is  equal  to  the  weight  divided  by  g.  And  g  is 
taken  equal  to  32-2;  but  the  acceleration  of  gravity  is  somewhat 
variable ;  it  becomes  greater  the  nearer  we  approach  the  poles  of 
the  earth.  It  is  greatest  at  the  poles  and  least  at  the  equator, 
and  also  diminishes  the  more  a  body  is  above  or  below  the  level  of 
the  sea.  The  mass,  so  long  as  nothing  is  added  to  or  taken  from 
it,  is  invariable,  whether  at  the  centre  of  the  earth  or  at  any  dis- 
tance from  it.  If  M  be  the  mass  and  W  the  weight  of  a  body, 

Then  M  =  -  =  3^2  =  -0310559  W. 

19.  What  is  the  mass  of  a  body  whose  weight  is  200  Ibs  ? 
•031055  x  200  =  6-21118  Ibs. 

The  weight  of  a  body  whose  mass  is  200  Ibs.  is  32-2  X  200  = 
6440-0  Ibs.  It  may  be  remarked,  that  one  and  the  same  steel 
spring  is  differently  bent  by  one  and  the  same  weight  at  different 
places. 

The  force  which  accelerates  the  motion  of  a  heavy  body  on  an 
inclined  plane,  is  to  the  force  of  gravity  as  the  sine  of  the  inclina- 


390  THE    PRACTICAL    MODEL    CALCULATOR. 

tion  of  the  plane  to  the  radius,  or  as  the  height  of  the  plane  to  its 
length. 

The  velocity  acquired  by  a  body  in  falling  from  rest  through  a 
given  height,  is  the  same,  whether  it  fall  freely,  or  descend  on  a 
plane  at  whatever  inclination. 

The  space  through  which  a  body  will  descend  on  an  inclined 
plane,  is  to  the  space  through  which  it  would  fall  freely  in  the  same 
time,  as  the  sine  of  the  inclination  of  the  plane  to  the  radius. 

The  velocities  which  bodies  acquire  by  descending  along  chords 
of  the  same  circle,  are  as  the  lengths  of  those  chords. 

If  the  body  descend  in  a  curve,  it  suffers  no  loss  of  velocity. 

The  centre  of  gravity  of  a  body  is  a  point  about  which  all  its 
parts  are  in  equilibrio. 

Hence,  if  a  body  be  suspended  or  supported  by  this  point,  the 
body  will  rest  in  any  position  into  which  it  is  put.  We  may,  there- 
fore, consider  the  whole  weight  of  a  body  as  centred  in  this  point. 

The  common  centre  of  gravity  of  two  or  more  bodies,  is  the  point 
about  which  they  would  equiponderate  or  rest  in  any  position.  If 
the  centres  of  gravity  of  two  bodies  be  connected  by  a  right  line, 
the  distances  from  the  common  centre  of  gravity  are  reciprocally 
as  the  weights  of  the  bodies. 

If  a  line  be  drawn  from  the  centre  of  gravity  of  a  body,  perpen- 
dicular to  the  horizon,  it  is  called  the  line  of  direction,  being  the 
line  that  the  centre  of  gravity  would  describe  if  the  body  fell  freely. 

The  centrS  of  gyration  is  that  part  of  a  body  revolving  about  an 
axis,  into  which  if  the  whole  quantity  of  matter  were  collected,  the 
same  moving  force  would  generate  the  same  angular  velocity. 

To  find  the  centre  of  G-yration. — Multiply  the  weight  of  the 
several  particles  by  the  squares  of  their  distances  from  the  centre 
of  motion,  and  divide  the  sum  of  the  products  by  the  weight  of  the 
whole  mass ;  the  square  root  of  the  quotient  will  be  the  distance 
of  the  centre  of  gyration  from  the  centre  of  motion. 

The  distances  of  the  centre  of  gyration  from  the  centre  of  mo- 
tion, in  different  revolving  bodies,  are  as  follow : — 

In  a  straight  rod  revolving  about  one  end,  the  length  X  '5773. 

In  a  circular  plate,  revolving  on  its  centre,  the  radius  X  *7071. 

In  a  circular  plate,  revolving  about  one  diameter,  the  radius  X  *5. 

In  a  thin  circular  ring,  revolving  about  one  diameter,  radius  X 
•7071. 

In  a  solid  sphere,  revolving  about  one  diameter,  the  radius  x 
•6325. 

In  a  thin  hollow  sphere,  revolving  about  one  diameter,  radius  X 
•8164. 

In  a  cone,  revolving  about  its  axis,  the  radius  of  the  base  X 
•5477. 

In  a  right-angled  cone,  revolving  about  its  vertex,  the  height  X 


SPECIFIC   GRAVITY.  391 

In  a  paraboloid,  revolving  about  its  axis,  the  radius  of  the  base 
x  -57T3. 

The  centre  of  percussion  is  that  point  in  a  body  revolving  about 
a  fixed  axis,  into  which  the  w hole  of  the  force  or  motion  is  collected. 

It  is,  therefore,  that  point  of  a  revolving  body  which  would  strike 
any  obstacle  with  the  greatest  effect ;  and,  from  this  property,  it 
has  received  the  name  of  the  centre  of  percussion. 

The  centres  of  oscillation  and  percussion  are  in  the  same  point. 

If  a  heavy  straight  bar,  of  uniform  density,  be  suspended  at  one 
extremity,  the  distance  of  its  centre  of  percussion  is  two-thirds  of 
its  length. 

In  a  long  slender  rod  of  a  cylindrical  or  prismatic  shape,  the 
centre  of  percussion  is  nearly  two-thirds  of  the  length  from  the 
axis  of  suspension. 

In  an  isosceles  triangle,  suspended  by  its  apex,  the  distance  of 
the  centre  of  percussion  is  three-fourths  of  its  altitude.  In  a  line 
or  rod  whose  density  varies  as  the  distance  from  the  point  of  sus- 
pension, also  in  a  fly-wheel,  and  in  wheels  in  general,  the  centre 
of  percussion  is  distant  from  the  centre  of  suspension  three-fourths 
of  the  length. 

In  a  very  slender  cone  or  pyramid,  vibrating  about  its  apex,  the 
distance  of  its  centre  of  percussion  is  nearly  four-fifths  of  its  length. 

Pendulums  of  the  same  length  vibrate  slower,  the  nearer  they 
are  brought  to  the  equator.  A  pendulum,  therefore,  to  vibrate 
seconds  at  the  equator,  must  be  somewhat  shorter  than  at  the  poles. 

When  we  consider  a  simple  pendulum  as  a  ball,  which  is  sus- 
pended by  a  rod  or  line,  supposed  to  be  inflexible,  and  without 
weight,  we  suppose  the  whole  weight  to  be  collected  in  the  centre 
of  gravity  of  the  ball.  But  when  a  pendulum  consists  of  a  ball, 
or  any  other  figure,  suspended  by  a  metallic  or  wooden  rod,  the 
length  of  the  pendulum  is  the  distance  from  the  point  of  suspension 
to  a  point  in  the  pendulum,  called  the  centre  of  oscillation,  which 
does  not  exactly  coincide  with  the  centre  of  gravity  of  the  ball. 

If  a  rod  of  iron  were  suspended,  and  made  to  vibrate,  that  point 
in  which  all  its  force  would  be  collected  is  called  its  centre  of  oscil- 
lation, and  is  situated  at  two-thirds  the  length  of  the  rod  from  the 
point  of  suspension. 


SPECIFIC  GRAVITY. 

THE  comparative  density  of  various  substances,  expressed  by  the 
term  specific  gravity,  affords  the  means  of  readily  determining  the 
bulk  from  the  known  weight,  or  the  weight  from  the  known  bulk ; 
and  this  will  be  found  more  especially  useful,  in  cases  where  the 
substance  is  too  large  to  admit  of  being  weighed,  or  too  irregular 
in  shape  to  allow  of  correct  measurement.  The  standard  with 
which  all  solids  and  liquids  are  thus  compared,  is  that  of  distilled 
water,  one  cubic  foot  of  which  weighs  1000  ounces  avoirdupois ; 


392  THE    PRACTICAL    MODEL    CALCULATOR.' 

and  the  specific  gravity  of  a  solid  body  is  determined  by  the  dif- 
ference between  its  weight  in  the  air,  and  in  water.     Thus, 

If  the  body  be  heavier  than  water,  it  will  displace  a  quantity  of 
fluid  equal  to  it  in  bulk,  and  will  lose  as  much  weight  on  immersion 
as  that  of  an  equal  bulk  of  the  fluid.  Let  it  be  weighed  first, 
therefore,  in  the  air,  and  then  in  water,  and  its  weight  in  the  air 
be  divided  by  the  difference  between  the  two  weights,  and  the  quo- 
tient will  be  its  specific  gravity,  that  of  water  being  unity. 

A  piece  of  copper  ore  weighs  56J  ounces  in  the  air,  and  43f 
ounces  in  water ;  required  its  specific  gravity. 
56-25  -  43-75  =  12-5  and  56-25  -v-  12-5  =  4-5,  the  specific  gravity. 

If  the  body  be  lighter  than  water,  it  will  float,  and  displace  a 
quantity  of  fluid  equal  to  it  in  weight,  the  bulk  of  which  will  be  equal 
to  that  only  of  the  part  immersed.  A  heavier  substance  must, 
therefore,  be  attached  to  it,  so  that  the  two  may  sink  in  the  fluid. 
Then,  the  weight  of  the  lighter  substance  in  the  air,  must  be  added 
to  that  of  the  heavier  substance  in  water,  and  the  weight  of  both 
united,  in  water,  be  subtracted  from  the  sum  ;  the  weight  of  the 
lighter  body  in  the  air  must  then  be  divided  by  the  difference,  and  the 
quotient  will  be  the  specific  gravity  of  the  lighter  substance  required. 

A  piece  of  fir  weighs  40  ounces  in  the  air,  and,  being  immersed 
in  water  attached  to  a  piece  of  iron  weighing  30  ounces,  the  two 
together  are  found  to  weigh  3-3  ounces  in  water,  and  the  iron  alone, 
25-8  ounces  in  the  water  ;  required  the  specific  gravity  of  the  wood. 

40  +  25-8  =  65-8  -  3-3  =  62-5;  and  40  -r-  62-5  =  0-64,  the 
specific  gravity  of  the  fir. 

The  specific  gravity  of  a  fluid  may  be  determined  by  taking  a 
solid  body,  heavy  enough  to  sink  in  the  fluid,  and  of  known  spe- 
cific gravity,  and  weighing  it  both  in  the  air  and  in  the  fluid.  The 
difference  between  the  two  weights  must  be  multiplied  by  the  spe- 
cific gravity  of  the  solid  body,  and  the  product  divided  by  the 
weight  of  the  solid  in  the  air :  the  quotient  will  be  the  specific 
gravity  of  the  fluid,  that  of  water  being  unity. 

Required  the  specific  gravity  of  a  given  mixture  of  muriatic  acid 
and  water ;  a  piece  of  glass,  the  specific  gravity  of  which  is  3, 
weighing  3{  ounces  when  immersed  in  it,  and  6  ounces  in  the  air. 

6  -  3-75  =  2-25  x  3  =  6-75  -r-  6  =  1-125,  the  specific  gravity. 

Since  the  weight  of  a  cubic  foot  of  distilled  water,  at  the  tem- 
perature of  60  degrees,  (Fahrenheit,)  has  been  ascertained  to  be 
1000  avoirdupois  ounces,  it  follows  that  the  specific  gravities  of  all 
bodies  compared  with  it,  may  be  made  to  express  the  weight,  in 
ounces,  of  a  cubic  foot  of  each,  by  multiplying  these  specific  gra- 
vities (compared  with  that  of  water  as  unity)  by  1000.  Thus,  that 
of  water  being  1,  and  that  of  silver,  as  compared  with  it,  being 
10-474,  the  multiplication  of  each  by  1000  will  give  1000  ounces 
for  the  cubic  foot  of  water,  and  10474  ounces  for  the  cubic  foot 
of  silver. 


SPECIFIC   GRAVITY.  393 

In  the  following  tables  of  specific  gravities,  the  numbers  in  the 
first  column,  if  taken  as  whole  numbers,  represent  the  weight  of  a 
cubic  foot  in  ounces  ;  but  if  the  last  three  figures  are  taken  as  deci- 
mals, they  indicate  the  specific  gravity  of  the  body,  water  being 
considered  as  unity,  or  1. 

To  ascertain  the  number  of  cubic  feet  in  a  substance,  from  its 
weight,  the  whole  weight  in  pounds  avoirdupois  must  be  divided  by 
the  figures  against  the  name,  in  the  second  column  of  the  table, 
taken  as  whole  numbers  and  decimals,  and  the  quotient  will  be  the 
contents  in  cubic  feet. 

Required  the  cubic  content  of  a  mass  of  cast-iron,  weighing  7  cwt. 
1  qr.  =  812  Ibs. 

812  Ibs.  -T-  450-5  (the  tabular  weight)  =  1-803  cubic  feet. 

To  find  the  weight  from  the  measurement  or  cubic  content  of  a 
substance,  this  operation  must  be  reversed,  and  the  number  of  cubic 
feet,  found  by  the  rules  given  under  "Mensuration  of  Solids," 
multiplied  by  the  figures  in  the  second  column,  to  obtain  the  weight 
in  pounds  avoirdupois. 

Required  the  weight  of  a  log  of  oak,  3  feet  by  2  feet  6  inches, 
and  9  feet  long. 

9x3x2-5  =  67-5  cubic  feet. 

And  67-5  X  58-2  (the  tabular  weight)  =  3928-5  Ibs.,  or  35  cwt. 
0  qr.  8J  Ibs. 

The'  velocity  g,  which  is  the  measure  of  the  force  of  gravity, 
varies  with  the  latitude  of  the  place,  and  with  its  altitude  above 
the  level  of  the  sea. 

The  force  of  gravity  at  the  latitude  of  45°  =  32-1803  feet ;  at 
any  other  latitude  L,  g  =  32-1803  feet  —  0-0821  cos.  2  L.  If 
gr  represents  the  force  of  gravity  at  the  height  h  above  the  sea, 
and  r  the  radius  of  the  earth,  the  force  of  gravity  at  the  level  of  the 

sea  will  be  g  =  gf  (I  +  ~). 

In  the  latitude  of  London,  at  the  level  of  the  sea,  g  =  32-191  feet. 
Do.  Washington,        do.          do.,    g  =  32-155  feet. 

The  length  of  a  pendulum  vibrating  seconds  is  in  a  constant 
ratio  to  the  force  of  gravity. 

|  =  9-8696044. 

Length  of  a  pendulum  vibrating  seconds  at  the  level  of  the  sea,  in 
various  latitudes. 

At  the  Equator 39-0152  inches. 

Washington,  lat.  38°  53'  23" 39-0958      — 

New  York,    lat.  40°  42'  40" 39-1017      — 

London,         lat.  51°  31' 39-1393      — 

lat.  45° 39-1270      — 

lat.i 39-1270  in.— 0-09982  cos.  2  L. 


394 


THE  PRACTICAL  MODEL  CALCULATOR. 


Specific  G-ravity  of  various  Substances. 


METALS. 

Antimony,  fused  . 
Bismuth,  cast   . 
Brass,  common,  cast    . 
cast         .... 
wire-drawn       .        . 
Copper,  cast 
wire-drawn       .        . 
Gold,  pure,  cast 
22  carats,  stand 
20  carats,  trinket  . 
Iron,  cast                      . 
bars        .... 
Lead,  cast     . 
litharge  .... 
Manganese                   . 
Mercury,  solid,  \ 
400  below  00  /• 
at  32  deg.  Fahr. 
at  60  deg. 
at  212  deg. 
Nickel,  cast       . 
Platina,  crude,  grains  . 
purified  .... 
hammered 
rolled     .... 
wire-drawn 
Silver,  cast,  pure     . 
Parisian  standard    . 
French  coin  .        .       . 
shilling,  Geo.  III.    . 
Steel,  soft  .... 
hardened   . 
tempered        .        .       . 
tempered  and  hard  • 
Tin,  pure  Cornish    . 
Tungsten      . 
Uranium    .... 
Wolfram       . 
Zinc,  usual  state 

Weight  of  a 
cubic  foot 
in  ounces. 

Weight  of  » 
cubic  foot 
in  pounds. 

STONES.  —  Continued. 

Grindstone        .... 
Gypsum,  opaque  . 
semi-transparent  . 
Jet,  bituminous     . 
Lime-stone       .... 
Marble   .        .        • 
Mill-stone         .... 
Porcelain,  China  . 
Portland-stone 
Pumice  stone        .       .       . 
Paving-stone    .... 
Purbeck-stone 

Weight  of  a 
cubic  foot 
in  ounces. 

Weight  of  a  ! 
cubic  foot 
in  pounds. 

6,624 
9,823 
7,824 
8,3% 
8,544 
8,788 
8,878 
19,258 
17,486 
15,709 
7,207 
7,788 
11,352 
6,300 
7,000 
15,632 

13,619 
13,580 
13,375 

7,807 
15,602 
19,500 
20,337 
22,069 
21.042 
10,474 
10,175 
10,048 
10,534 
7,833 
7,840 
7,816 
7,818 
7,291 
6,066 
6,440 
7,119 
6,862 
7,191 

845 
852 
912 
1,328 
1,031 
561 
1,315 
715 
1,040 
240 
1,209 
1,331 
671 
657 
569 
1,333 
604 
913 
1,063 
750 
1,170 
932 
1,327 
671 
585 
807 

2,699 
2,730 
1,714 
2,000 
2,784 
1,270 
1,270 
1,240 
1,300 
4,000 
2,582 
2,933 
2,892 
2,625 
2,662 
2,654 
2,728 

414-0 
614-0 
489-0 
524-8 
534-0 
549-2 
554-9 
1203-6 
1093-0 
982-0 
450-5 
486-8 
709-5 
393-8 
437-5 

977-0 
851-2 
848-8 
836-8 
488-0 
975-1 
1218-8 
1271-1 
1379-4 
1315-1 
654-6* 
636-0 
628-0 
658-4 
489-6 
490-0 
488-5 
488-6 
455-6 
379-1 
402-5 
445-0 
429-0 
449-5 

52-9 

53-2 
57-0 
83-0 
64-5 
35-1 
82-2 
44-8 
65-0 
15-0 
75-6 
83-2 
42-0 
41-1 
35-6 
83-4 
37-8 
57-1 
66-5 
47-0 
7M 
58-2 
83-0 
42-0 
36-6 
60-5 

168-8 
170-6 
107-1 
125-0 
174-0 
79-4 
79-4 
77-5 
81-2 
250-0 
162-0 
170-9 
168-2 
164-1 
166-4 
165-9 
170-5 

2,143 
2,168 
2,306 
1,259 
3,182 
2,700 
2,484 
2,385 
2,570 
915 
2,416 
2,601 
1,981 

134-0 
135-5 
144-1 
78-8 
199-0 
168-8 
155-2 
149-1 
160-6 
57-2 
151-0 
162-6 
124-0 
167-0 
178-4 
157-5 
154-4 
127-1 
124-5 

63-0 
64-1 
52-3 

51-8 
56-1 
63-0 
46-2 
64-5 
74-6 
79-5 
99-0 
57-4 
58-0 
58-8 
57-3 
57-8 
57-5 
54-4 
57-8 
52-4 

51-9 
115-1 
133-0 
62-0 
63-1 
62-5 
64-1 

60-4 
59-0 
62-0 
57-8 
58-6 
90-6 
48-1 
114-1 
59-2 
83-5 
59-0 
100-4 
59-0 

1-000 
•590 
•969 
1-520 
•960 
•491 
•470 
•074 
1-278 
1-094 
2-427 
1-104 
•690 
1-777 
2-193 

Slate,"  common 
new        

Stone,  common     .        . 

2,672 
2,854 
2,520 
2,470 
2,033 
1,991 

1007 

Sulphur,  native    .       . 

LIQUIDS. 
Acetic  acid       .... 

Acetous  acid          .        . 
Alcohol,  commercial 
highly  rectified  . 
Ammonia,  liquid      . 
Beer       
Ether,  sulphuric 
Milk  of  cows         .        .        . 
Muriatic  acid    .... 
Nitric  acid     .... 
highly  concentrated 
Oil  of  almonds,  sweet  . 
hemp-seed     .... 
linseed        .... 
olives     
poppies       .... 
rape-seed       .... 
turpentine,  essence  . 
whales    
Spirits  of  wine,  \ 
commercial  )  * 
highly  rectified     . 
Sulphuric  acid 
highly  concentrated     . 
Turpentine,  liquid 
Vinegar,  distilled    . 
Water,  rain,  or  distilled 

1,025 
837 
829 
897 
1,023 
739 
1,032 
1,194 
1,271 
1,583 
917 
926 
940 
915 
924 
919 
870 
923 

837 
829 
1,841 
2,125 
991 
1,010 
1,000 
1,026 

WOODS. 
Ash 

Beech    

Box,  Dutch 
French                       . 
Brazilian 
Cedar,  American  .       . 
Indian    .... 
Cherry-tree  . 
Cocoa         .... 
Cork              . 
Ebony,  Indian  . 
American 
Elm    
Fir,  yellow 
white      .... 
Lignum-vitse 
Lime-tree. 
Logwood                       . 
Mahogany 
Maple           .        .       .       . 
Oak,  heart  of,  old    . 
dry     
Vine  
Walnut         . 
Willow       .... 
Yew      .       .       .       .       . 

STONES,  EARTHS,  ETC. 

Alabaster,  yellow 
white      .... 
Borax    
Brick  earth 
Chalk    
Coal,  Cannel     . 
Newcastle 
Staffordshire 
Scotch        . 
Emery  
Flint,  black.        .        .        . 
Glass,  flint 
white         . 
Granite,  Aberd.  blue 
Cornish     .... 
Egyptian,  red 
gray 

MISCELLANEOUS   SUB- 
STANCES. 

965 
942 
989 
923 
937 
1,450 
769 
1,826 
948 
1,336 
943 
1,606 
942 

itimated 

Butter 

Fat,  beef  or  mutton     . 
hogs'       

T    f\' 

...         ° 

Lard                          .        . 

Spermaceti        .... 
Sugar,  white  .... 
Tallow 

GASES. 
'  Atmospheric  air  being  e 
as  1. 

Atmospheric,  or  common  air 
Ammoniacal  gas 
Azote    
Carbonic  acid    .... 

Carbonic  oxide 
Carburetted  hydrogen     . 

Steam    
Sulphuretted  hydrogen   .... 
Sulphurous  acid  

SPECIFIC    GRAVITY. 


395 


TABLE  of  the  Weight  of  a  Foot  in  length  of  Flat  and  Rolled  Iron. 


i 

BREADTH   IN  INCHES  AND  PARTS  OF  AN  INCH. 

4 

3f 

«i 

8* 

3 

2| 

2i 

^ 

2       If 

it 

It 

it 

1 

J_ 

* 

i 

1-68 

1-57 

1-47 

1-36 

1-26 

1-15 

1-05 

0-94 

0-84   0-73 

0-63 

0-57 

0-52 

0-42 

0-31 

0-21 

T3* 

2-52 

2-36 

2-20 

2-04 

1-89 

1-73 

1-57 

1-41 

1-26 

1-10 

0-94 

0-86 

0-78 

0-6o 

0-47 

0-31 

* 

3-36 

3-15 

2-94 

2-73 

2-52 

2-31 

2-10 

1-89 

1-68 

1-47 

1-26 

1-18 

1-05 

0-84 

0-63 

0-42 

1 

5-04 

4-72 

4-41 

4-09 

3-78 

3-46 

3-15 

2-83 

2-52 

2-20 

1-89 

1-73 

1-57 

1-20 

0-94 

0-63 

i 

6-72 

6-30 

5-88 

6-46 

5-04 

4-62 

4-20 

3-78 

3-36 

2-94 

2-52 

2-31 

2-10 

1-68 

1-26 

i 

8-40 

7-87 

7-35 

6-82 

6-30 

5-77 

5-25 

4-72 

4-20 

3-67 

3-15 

2-88 

2-62 

2-10 

1-57 

* 

10-08 

9-45 

8-82 

8-19 

7-56 

6-93 

6-30 

5-66 

5-04 

4-41 

3-78 

3-46 

3-15 

2-52 

7 

TT 

11-76 

11-02 

10-29 

9-45 

8-82 

8-08 

7-35 

6-61 

5-88 

5-14 

4-41 

4-04 

3-67 

2-94 

1 

13-44 

12-60 

11-76 

10-92 

10-08 

9-24 

8-40 

7-56 

6-72 

5-87 

5-04 

4-62 

4-20 

H 

15-12 

14-16 

13-20 

12-28 

11-34 

10-39 

9-45 

8-50 

7-56 

6-60 

5-67 

5-19 

4-72 

H 

16-80 

15-75 

14-70 

13-65 

12-60 

11-55 

10-50 

9-45 

8-40 

7-35 

6-30 

5-77 

if 

18-48 

17-32 

16-16 

15-01 

13-86 

12-70 

11-55 

10-39 

9-24 

8-07 

1* 

20-18 

18-90 

17-64 

16-38 

15-12 

13-86 

12-60 

11-34 

10-08 

8-80 

1* 

23-54 

22-05 

20-58 

19-11 

17-64 

16-17 

14-70 

13-22 

2 

26-88 

25-20 

23-52 

21-84 

20-16 

18-48 

16-80 

15-12 

2* 

33-65 

31-50 

29-40 

27-39 

25-20 

23-10 

3 

40-32 

37-80 

35-28 

32-76 

si 

47-04 

TABLE  of  the  Weight  of  Oast-iron  Pipes,  in  lengths. 


0 

i 

1 

A 

Weight. 

i 
I 

1 

H 

I 

Weight. 

I 

H 

§ 

Weight. 
i 

Inch. 

Inch. 

Feet. 

C.  qr.  Ib. 

Inch. 

Inch. 

Feet. 

C.  qr.  Ib. 

Inch. 

Inch. 

Feet. 

C.  qr.  Ib. 

1 

i 

3* 

12 

6* 

g 

9 

2  0  16 

11* 

^ 

9 

507 

f 

3* 

21 

£ 

9 

2  3  20 

s 

9 

6  1  12 

l* 

i 

4* 

21 

f 

9 

3  2  21 

i 

9 

728 

1 

4* 

1     4 

1 

9 

4  1  21 

1 

9 

10   1     2 

2 

a 

6 

1     8 

1 

9 

6  0  14 

12 

9 

5  0  24 

i 

6 

2     0 

7 

* 

9 

307 

I 

9 

628 

H 

i 

6 

1  16 

» 

9 

3  3  20 

9 

7  3  20 

t 

6 

2   10 

1 

9 

435 

1 

9 

10  3     0 

* 

6 

3  10 

1 

9 

624 

12* 

9 

511-6 

3 

i 

9 

2  20 

fi 

* 

9 

316 

i 

9 

639 

t 

9 

106 

i 

9 

4  0  22 

1 

9 

810 

* 

9 

1  1  12 

i 

9 

5  0  10 

9 

11  0  21 

$ 

9 

136 

i 

9 

700 

13 

1 

9 

5  2  20 

i 

9 

210 

8 

i 

9 

324 

K 

9 

7  0  14 

Si 

i 

9 

3     0 

9 

4  1  25 

3 

9 

827 

i 

9 

1  0  21 

1 

9 

5  1  18 

I 

9 

11   2  12 

i 

9 

1  2  14 

i 

9 

7  1  16 

13* 

£ 

9 

537 

& 

9 

208 

8* 

* 

9 

332 

$ 

9 

7  1  12 

1 

9 

220 

£ 

9 

4  2  26 

| 

9 

8  3  16 

4 

f 

9 

1  1  10 

£ 

9 

5  2  22 

9 

11  3  24 

* 

9 

1  3  12 

1 

9 

738 

14 

* 

9 

604 

1 

9 

2  1  12 

9 

* 

9 

400 

i 

9 

7  2  16 

1 

9 

2  2  21 

1 

9 

504 

9 

910 

4* 

§ 

9 

122 

£ 

9 

602 

1 

9 

12  1  -14 

* 

9 

204 

1 

9 

8  0  26 

14* 

9 

6  0  24 

£ 

9 

2  2  14 

9* 

£ 

9 

4  0  18 

f 

9 

7  3  14 

1 

9 

3  0  21 

t 

9 

510 

9 

922 

5 

I 

9 

1  2  22 

1 

9 

616 

J 

9 

12  3     6 

i 

9 

2  1  10 

1 

9 

8  2  20 

15 

1 

9 

6  1  21 

f 

9 

2  3  17 

10 

* 

9 

4  1   10 

s 

9 

937 

H 

1 

9 
9 

3  1  24 
1  3  10 

9 
9 

5   1  26 

4  2  14 

9 
9 

13  0  26 
16  3     5 

i 

9 

220 

1 

9 

908 

15* 

^ 

9 

6  2  14 

£ 

9 

3  0  18 

10* 

* 

9 

4  2  14 

i 

9 

10  9  10 

f 

9 

337 

§ 

9 

537 

9 

13  2  17 

1 

9 

5  0  12 

1 

9 

700 

li 

9 

17  1     6 

6 

§ 

9 

200 

1 

9 

920 

16 

* 

9 

7  0  22 

* 

9 

2  2  21 

11 

* 

9 

4  3  14 

9 

10  1  20 

f 

9 

3  1  17 

§ 

9 

6  0  11 

j 

9 

14  0     8 

1 

9 

4  0  16 

1 

9 

7  1     7 

li 

9 

17  3  14 

1 

9 

5  2  20 

1 

9 

9  3  20 

H 

9 

21  3     4 

396 


THE    PRACTICAL    MODEL    CALCULATOR. 


TABLE  of  the  Weight  of  one  Foot  Length  of  Malleable  Iron. 


SQUARE   IRON. 

ROU-VD  IRON. 

Scantling. 

Weight. 

Diameter. 

Weight. 

Circumference.  1         Weight. 

Inches. 

Pounds. 

Inches. 

Pounds. 

laches. 

Pounds. 

0-21 

I 

0-16 

1 

0-26 

0-47 

4 

0-37 

u 

0-41 

0-84 

| 

0-66 

11 

0-59 

1-34 

-| 

1-03 

If 

0-82 

1-89 

| 

1-48 

2 

1-05 

\ 

~ 

2-57 

2-02 

*l 

1-34 

1 

3-36 

1¥ 

2-63 

2£ 

1-65 

*1 

4-25 

H 

3-33 

2f 

2-01 

1 

5-25 

if 

4-12 

3 

2-37 

11 

6-35 

if 

4-98 

8J 

2-79 

lj 

7-56 

i| 

5-93 

3£ 

3-24 

1- 

8-87 

if 

6-96 

3f 

3-69 

1| 

10-29 

if 

8-08 

4 

4-23 

1, 

11-81 

» 

9-27 

4£ 

5-35 

2 

13-44 

2 

10-55 

5 

6-61 

2* 

2* 

17-01 
21-00 

It 

13-35 
16-48 

^ 
6 

7-99 
9-51 

2f 

25-41 

2f 

19-95 

H 

11-18 

3 

30-24 

3 

23-73 

7 

12-96 

3£ 

41-16 

3i 

27-85 

7| 

14-78 

4 

53-76 

3} 

32-32 

8 

16-92 

4* 

68-04 

3| 

37-09 

3| 

19-21 

6 

84-00 

4 

42-21 

9 

21-53 

6 

120-96 

4£ 

53-41 

10 

26-43 

7 

164-64 

5 

65-93 

12 

31-99 

The  following  tables  are  rendered  of  great  utility  by  means  of 
this  table : — 


The  weight  of  Water 

being 

1- 
8-8 
8-4 
7-2 
11-3 
7-2 
8-7 
1-5 
1-25 
2-0 
2-5 
0-85 

= 

Lead 

Zinc 

= 

Sand 

= 

Brick 

Stone 

age  = 

Suppose  it  be  required  to  ascertain  the  weight  of  a  cast  iron 
pipe  26J  inches  outside  and  23|  inside,  the  length  being  6J  feet. 
Opposite  26J  in  the  table  is 

234-8576  x  7-2  x  6-5  =  10991-135. 
And  opposite  23}  in  the  table  is 

192-2856  x  7-2  X  6-5  =  8998-966  subtract 

1992-169  Ibs.  avr. 

The  succeeding  table  contains  the  surface  and  solidity  of  spheres, 
together  with  the  edge  or  dimensions  of  equal  cubes,  the  length 
of  equal  cylinders,  and  the  weight  of  water  in  avoirdupois  pounds : — 


SPECIFIC    GRAVITY. 


Surface  and  Solidity  of  Spheres. 


Diameter. 

Surface. 

Solidity. 

Cube. 

Cylinder. 

Water  in  Ibs. 

lin. 

3-1416 

•5236 

•8060 

•6666 

•0190 

ft 

3-5465 

,      -6280 

•8563 

•7082 

•0227 

i 

3-9760 

•7455 

•9067 

•7500 

•0270 

& 

4-4301 

•8767 

•9571 

•7917 

•0317 

\ 

4-9087 

1-0226 

1-0075 

•8333 

•0370 

& 

5-4117 

1-1838 

1-0578 

•8750 

•0428 

f 

5-9395 

1-3611 

1-1082 

•9166 

•0500 

A 

6-4918 

1-5553 

1-1586 

•9583 

•0563 

i 

7-0686 

1-7671 

1-2090 

1-0000 

•0640 

A 

7-6699 

2-0000 

1-2593 

1-0416 

•0723 

i 

8-2957 

2-2467 

1-3097 

1-0833 

•0813, 

ft 

8-9461 

2-5161 

1-3601 

1-1349 

•0910 

t 

9-6211 

2-8061 

1-4105 

1-1666 

•1015 

it 

10-3206 

3-1176 

1-4608 

1-2083 

•1128 

i 

11-0446 

3-4514 

1-5112 

1-2500 

•1250 

if 

11-7932 

3-8081 

1-5616 

1-2916 

•1377 

2  in. 

12-5664 

4-1888 

1-6020 

1-3333 

•1516 

ft 

13-3640 

4-5938 

1-6633 

1-3750 

•1662 

i 

14-1862 

5-0243 

1-7127 

1-4166 

•1818 

ft 

15-0330 

5-4807 

1-7631 

1-4582 

•1982 

i 

15-9043 

6-9640 

1-8135 

1-5000 

•2160 

T5* 

16-8000 

6-4749 

1-8638 

1-5516 

•2342 

1 

17-7205 

7-0143 

1-9142 

1-5832 

•2540 

ft 

18-6655 

7-5828 

1-9646 

1-6250 

•2743 

| 

19-6350 

8-1812 

2-0150 

1-6666 

•2960 

T9* 

20-6290 

8-8103 

2-0653 

1-7082 

•3187 

1 

21-6475 

9-4708 

2-1157 

1-7500 

•3426 

tt 

22-6907 

10-1634 

2-1661 

1-7915 

•3676 

1 

23-7583 

10-8892 

2-2165 

1-8332 

•3939 

H 

24-8505 

11-6485 

2-2668 

1-8750 

•4213 

I 

25-9672 

12-4426 

2-3172 

1-9165 

•4501 

If 

27-1084 

13-2718 

2-3676 

1-9582 

•4800 

3  in. 

28-2744 

14-1372 

2-4180 

2-0000 

•5114 

ft 

29-4647 

15-0392 

2-4683 

2-0415 

•5440 

i 

30-6796 

15-9790 

2-5187 

2-0832 

•5780 

T3* 

31-9191 

16-9570 

2-5691 

2-1250 

•6133 

i 

33-1831 

17-9742 

2-6195 

2-1665 

•6401 

T5S 

35-3715 

19-0311 

2-6698 

2-2082 

•6884 

I 

35-7847 

20-1289 

2-7202 

2-2500 

•7281 

§ 

37-1224 

21-2680 

2-7706 

2-2915 

•7693 

i 

38-4846 

22-4493 

2-8210 

2-3332 

•8120 

T9* 

39-8713 

23-6735 

2-8713 

2-3750 

•8561 

1 

41-2825 

24-9415 

2-9217 

2-4166 

•9021 

I* 

42-7183 

26-2539 

2-9712 

2-4582 

•9496 

I 

44-1787 

27-6117 

3-0225 

2-5000 

•9987 

It 

45-6636 

29-0102 

3-0728 

2-5415 

1-0493 

I 

47-1730 

30-4659 

3-1232 

2-5832 

1-1020 

If 

48-7070 

31-9640 

3-1730 

2-6250 

1-1561 

4  in. 

50-2656 

33-5104 

3-2240 

2-6665 

1-1974 

ft 

51-8486 

35-1058 

3-2743 

2-7082 

1-2698 

i 

53-4562 

36-7511 

3-3247 

2-7500 

1-3293 

T3* 

55-0884 

38-4471 

3-3751 

2-7915 

1-3906 

i 

56-7451 

40-1944 

3-4255 

2-8332 

1-4538 

ft 

58-4262 

42-0461 

3-4758 

2-8750 

1-5208 

1 

60-1321 

43-8463 

3-5262 

2-9165 

1-5860 

§ 

61-8625 

45-7524 

3-5766 

2-9582 

1-6550 

398 


THE    PRACTICAL    MODEL    CALCULATOR. 


Diameter. 

Surface. 

Solidity. 

Cube. 

Cylinder. 

Water  in  Ibs. 

| 

63-6174 

47-7127 

3-6270 

3-0000 

1-7258 

l9* 

65-3968 

49-7290 

3-6773 

3-0415 

1-7987 

1 

67-2007 

51-8006 

3-7277 

3-0832 

1-8736 

H 

69-0352 

53-9290 

3-7781 

3-1250 

1-9506 

I 

70-8823 

56-1151 

3-8285 

3-1665 

2-0297 

ft 

72-7599 

58-3595 

3-8788 

3-2080 

2-1109 

7 

¥ 

74-6620 

60-6629 

3-9292 

3-2500 

2-1942 

if 

76-5887 

62-9261 

3-9796 

3-2913 

2-2760 

5  m. 

78-5400 

65-4500 

4-0300 

3-3332 

2-3673 

iV 

80-5157 

67-9351 

4-0803 

3-3750 

2-4572 

{ 

82-5160 

70-4824 

4-1307 

3-4155 

2-5453 

A 

84-5409 

73-0926 

4-1811 

3-4582 

2-6438 

1 

86-5903 

75-7664 

4-2315 

3-5000 

2-7605 

& 

88-6641 

78-5077 

4-2818 

3-5414 

2-8396 

1 

90-7627 

81-3083 

4-3322 

3-5832 

2-9407 

A 

92-8858 

84-1777 

4-3820 

3-6250 

3-0447 

1 

95-0334 

87-1139 

4-4330 

3-6665 

3-1509 

& 

•  97-2053 

90-1175 

4-4633 

3-7080 

3-2595 

1 

99-4021 

93-1875 

4-5337 

3-7500 

3-3706 

U 

101-6233 

96-3304 

4-5841 

3-7913 

3-4843 

3. 

4 

103-8691 

99-5412 

4-6345 

3-8330 

3-6004 

3 

106-1394 

102-8225 

4-6848 

3-8750 

3-7191 

j 

108-4342 

106-1754 

4-7352 

3-9163 

3-8404 

it 

110-7536 

109-5973 

4-7856 

3-9580 

3-9641 

6  in. 

113-0976 

113-0976 

4-8360 

4-0000 

4-0907 

£ 

115-4660 

116-6688 

4-8863 

4-0417 

4-2200 

? 

117-8590 

120-3139 

4-9367 

4-0833 

4-3517 

T3S 

120-2771 

124-0374 

4-9871 

4-1250 

4-4874 

i 

122-7187 

127-8320 

5-0375 

4-1666 

4-6236 

T55 

125-1852 

131-7053 

5-0878 

4-2083 

4-7638 

i 

127-6765 

135-6563 

5-1382 

4-2500 

4-9067 

A 

130-1923 

139-6854 

5-1886 

4-2917 

5-0524  ' 

| 

132-7326 

143-7936 

5-2390 

4-3332 

5-2010 

T9S 

135-2974 

147-9815 

5-2893 

4-3750 

5-3525 

137-8867 

152-2499 

5-3377 

4-4165 

5-5069 

ii 

140-5006 

156-5997 

5-3901 

4-4583 

5-6786 

? 

143-1391 

161-0315 

5-4405 

4-5000 

5-8245 

if 

145-8021 

167-5461 

5-4908 

4-5416 

6-0601 

f 

148-4896 

170-1682 

5-5412 

4-5832 

6-1550 

if 

151-2017 

174-8270 

5-5916 

4-6250 

6-3235 

w  > 
7  m. 

153-9384 

179-5948 

5-6420 

4-6665 

6-4960 

X 

156-6995 

184-4484 

5-6923 

4-7082 

6-6725 

I 

159-4852 

189-3882 

5-7427 

4-7500 

6-8502 

T3S 

162-2955 

194-1165 

5-7931 

4-7915 

7-0212 

? 

165-1303 

199-5325 

5-8435 

4-8332 

7-2171 

X 

167-9895 

204-7371 

5-8938 

4-8750 

7-4053 

1 

170-8735 

210-0331 

5-9442 

4-9166 

7-5970 

T\ 

173-7520 

215-4172 

5-9946 

4-9582 

7-7916 

Y 

176-7150 

220-8937 

6-0450 

5-0000 

7-9897 

T9ff 

179-6725 

226-7240 

6-0953 

5-0415 

8-2006 

1 

182-6545 

232-1235 

6-1457 

5-0832 

8-3960 

M 

185-6611 

237-8883 

6-1961 

5-1250 

8-6044 

I 

188-6923 

243-7276 

6-2465 

5-1665 

8-8157 

12- 

191-7480 

249-4720 

6-2968 

5-2082 

9-0234 

1 

194-8282 

255-7121 

6-3472 

5-2500 

9-2491 

o 

4-4 

197-9330 

261-9673 

6-3976 

5-2913 

9-4753 

1  6  • 

8  in. 

201-0624 

268-0832 

6-4480 

5-3330 

9-6965 

TV 

204-2162 

274-4156 

6-4983 

5-3750 

9-9260 

SPECIFIC    GRAVITY. 


399 


Diameter. 

Surface. 

Solidity. 

Cube. 

Cylinder. 

Water  iu  Its. 

| 

207-3946 

280-8469 

6-5487 

5-4164 

10-1583 

T3S 

210-5976 

287-3780 

6-5991 

5-4581 

10-3944 

i 

213-8251 

294-0095 

6-6495 

5-5000 

10-6343 

T5* 

217-0770 

300-7422 

6-6998 

5-5414 

10-8778 

1 

220-3537 

307-5771 

6-7502 

5-5831 

11-1250 

A 

223-6549 

314-5147 

6-8006 

5-6250 

11-3760 

* 

226-9806 

321-5553 

6-8510 

5-6664 

11-6306 

A 

230-3308 

328-7012 

6-9013 

5-7080 

11-8891 

i 

233-7055 

335-9517 

6-9517 

5-7500 

12-1514 

u 

237-1048 

343-3079 

7-0021 

5-7913 

12-4170 

i 

240-5287 

350-7710 

7-0525 

5-8330 

12-6874 

ft 

243-9771 

358-3412 

7-1028 

5-8750 

12-9612 

i 

247-4500 

366-0199 

7-1532 

5-9163 

13-2390 

it 

250-9475 

373-8073 

7-2036 

5-9580 

13-5206 

9  in. 

254-4696 

381-7017 

7-2540 

6-0000 

13-8062 

T'e 

258-0261 

389-7118 

7-3043 

6-0417 

14-0959 

i 

261-5872 

397-8306 

7-3547 

6-0833 

14-3895 

A 

265-1829 

406-0613 

.  7-4051 

6-1250 

14-6872 

i 

268-8031 

414-4048 

_  7-4555 

6-1667 

14-9890 

A 

272-4477 

421-2907 

7-5058 

6-2083 

15-2381 

t 

276-1171 

431-4361 

7-5562 

6-2500 

15-6050 

TV 

279-8110 

440-1294 

7-6066 

6-2916 

15-9195 

? 

283-5294 

448-9215 

7-6570 

6-3333 

16-2375 

T9* 

287-2723 

457-8500 

7-7073 

6-3750 

16-5604 

1 

291-0397 

466-8763 

7-7557 

6-4166 

16-6869 

fl 

294-8310 

476-0304 

7-8081 

6-4582 

17-2180 

1 

298-4483 

485-3035 

7-8585 

6-5000 

17-5534 

ft 

302-4894 

494-6952 

7-9088 

6-5415 

17-8931 

* 

306-3550 

504-2094 

7-9592 

6-5832 

18-2373 

if 

310-9452 

513-8436 

8-0096 

6-6250 

18-5857 

10  in. 

314-1600 

523-6000 

8-0600 

6-6666 

18-6786 

iV 

318-0992 

533-4789 

8-1103 

6-7083 

19-2960 

i 

322-0630 

543-4814 

8-1607 

6-7500 

19-6577 

T3ff 

326-0514 

553-6081 

8-2111 

6-7916 

20-0240 

i 

330-0643 

563-8603 

8-2615 

6-8333 

20-3948 

A 

334-1016 

574-2371 

8-3118 

6-8750 

20-6682  . 

1 

338-1637 

584-7415 

8-3622 

6-9166 

21-1501 

& 

342-2503 

595-3677 

8-4126 

6-9582 

21-5344 

f 

346-3614 

606-1318 

8-4630 

7-0000 

21-9238 

T9S 

350-4970 

617-0207 

8-5133 

7-0416 

22-3176 

1 

354-6571 

628-0387 

8-5637 

7-0833 

22-7162 

H 

358-8418 

639-1871 

8-6141 

7-1250 

23-1194 

¥ 

363-0511 

650-4666 

8-6645 

7*1666 

23-5274 

if 

367-2849 

661-8580 

8-7148 

7-2082 

23-9394 

i 

371-5432 

673-4222  ' 

8-7652 

7-2500 

24-3577 

if 

375-8261 

685-0997 

8-8156 

7-2915 

24-7801 

11  in. 

380-1336 

696-9116 

8-8660 

7-3330 

25-2073 

A 

384-4655 

708-9106 

8-9163 

7-3750 

25-6414 

s 

388-8220 

720-9409 

8-9667 

7-4165 

26-0764 

A 

393-2031 

733-1599 

9-0171 

7-4582 

26-5184 

* 

397-6087 

745-5004 

9-0675 

7-5000 

26-5657 

T5* 

402-0387 

758-0104 

9-1178 

7-5414 

27-4162 

1 

406-4935 

770-6440 

9-1682 

7-5832 

27-8742 

Tv 

410-7728 

783-5787 

9-2186 

7-6250 

28-3420 

? 

415-4766 

796-3301 

9-2690 

7-6664 

28-8033 

T9S 

420-0049 

809-3844 

9-3193 

7-7080 

29-2754 

i 

424-5576 

822-5807 

9-3697 

7-7500 

29-7527 

H 

429-1351 

835-9695 

9-4201 

7-7913 

30-2370 

400 


THE   PRACTICAL   MODEL   CALCULATOR. 


Diameter. 

Surface. 

Solidity. 

Cube. 

Cylinder. 

Water  in  Ibs. 

I 

4337371 

849-4035 

9-4705 

7-8330 

30-7229 

II 

438-3636 

863-0283 

9-5208 

7-8750 

31-2157 

f 

443-0146 

876-7999 

9-5772 

7-9163 

31-3883 

If 

447-6902 

890-7070 

9-6216 

7-9580 

32-2169 

12  in. 

452-3904 

904-7808 

9-6720 

8-0000 

32-7259 

| 

471-4363 

962-5158 

9-8735 

8-1666 

34-8142 

490-8750 

1022-656 

10-0750 

8-3332 

36-9886 

| 

506-7064 

1085-251 

10-2765 

8-5000 

39-2535 

13  in. 

530-9304 

1150-337 

10-4780 

8-6666 

41-6077 

| 

551-5471 

1218-000 

10-6790 

8-8332 

44-0551 

1 

572-5566 

1288-252 

10-8810 

9-0000 

46-5961 

f 

593-9587 

1361-346 

11-0825 

9-1665 

49-239'J 

14  in. 

615-7536 

1436-758 

11-2840 

9-8332 

51-9675 

i 

637-9411 

1515-106 

11-4855 

9-5000 

54-8014 

i 

660-5214 

1596-260 

11-6870 

9-6665 

57-7367 

f 

683-4943 

1680-265 

11-8885 

9-8332 

60-7751 

15  in. 

706-8600 

1767-150 

12-0900 

10-0000 

64-0178 

fc 

730-6183 

1856-988  - 

12-2915 

10-1666 

67-1672 

754-7694 

1949-821 

12-4930 

10-3332 

70-5250 

f 

779-3131 

2045-697 

12-6940 

10-5000 

73-9929 

16  in. 

804-2496 

2144-665 

12-8960 

10-6666 

77-5725 

TABLE  containing  the  Weight  of  Flat  Bar  Iron,  1  foot  in  length, 
of  various  breadths  and' thicknesses. 


fl 

2  A 

THICKNESS  IN  PARTS  OF  AN  INCH. 

\ 

A 

f 

iV 

* 

T9* 

1 

f 

I 

1  inch. 

«M 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1  in. 

0-83 

1-04 

1-25 

1-45 

1-66 

1-87 

2-08 

2-50 

2-91 

3-33 

Ij 

0-93 

1-17 

140 

1-64 

1-87 

2-00 

2-34 

2-81 

3-28 

3-75 

1; 

1-04 

1-30 

1-56 

1-82 

2-08 

2-34 

2-60 

3-12 

3-74 

4-16 

If 

1-14 

143 

1-71 

2-00 

2-29 

2-57 

2-86 

343 

4-01 

4-58 

li 

1-25 

1-56 

1-87 

2-18 

2-50 

2-81 

3-12 

3-75 

4-37 

5-00 

iji 

1-35 

1-69 

2-03 

2-36 

2-70 

3-04 

3-38 

4-06 

4-73 

541 

i| 

1-45 

1-82 

2-18 

2-55 

2-91 

3-28 

3-64 

4-37 

5-10 

5-83 

1} 

1-56 

1-95 

2-34 

2-73 

3-12 

3-51 

3-90 

4-68 

546 

6-25 

2  in. 

1-66 

2-08 

2-50 

2-91 

3-33 

3-75 

4-16 

5-00 

5-83 

6-6G 

2i 

1-77 

2-21 

2-65 

3-09 

3-54 

3-98 

442 

5-31 

6-19 

7-08 

2$ 

1-87 

2-34 

2-81 

3-28 

3-75 

4-21 

4-68 

5-62 

6-56 

7-50 

2I 

1-97 

2-47 

2-96 

3-46 

3-95 

445 

4-94 

5-93 

6-92 

7-91 

2* 

2-08 

2-60 

3-12 

3-64 

4-16 

4-68 

5-20 

6-25 

7-29 

8-33 

2£ 

2-18 

2-73 

3-28 

3-82 

4-37 

4-92 

546 

6-56 

7-65 

8-75 

2f 

2-29 

2-86 

3-43 

4-01 

4-58 

5-15 

5-72 

6-87 

8-02 

9-16 

2f 

2-39 

2-99 

3-59 

4-19 

4-79 

5-39 

5-98 

7-18 

8-38 

9-58 

3  in. 

2-50 

3-12 

3-75 

4-37 

5-00 

5-62 

6-25 

7-50 

8-75 

10-00 

3i- 

2-70 

3-38 

4-06 

4-73 

541 

6-09 

6-77 

8-12 

947 

10-83 

3} 

2-91 

3-64 

4-37 

5-10 

5-83 

6-56 

7-29 

8-75 

10-20 

11-66 

3f 

3-12 

3-90 

4-68 

5-46 

6-25 

7-03 

7-81 

9-37 

10-93 

12-50 

4  in. 

3-33 

4-16 

5-00 

5-83 

6-66 

7-50 

8-33 

10-00 

11-66 

13-33 

4| 

3-54 

442 

5-31 

6-19 

7-08 

7-96 

8-85 

10-62 

12-39 

14-16 

3-75 

4-68 

5-62 

6-56 

7-50 

843 

9-37 

11-25 

13-12 

15-00 

4f 

3-95 

4-94 

5-93 

6-92 

7-91 

8-90 

9-89 

11-87 

13-85 

15-33 

5  in. 

4-17 

5-20 

6-25 

7-29 

8-33 

9-37 

1041 

12-50 

14-58 

16-66 

5i 

4-37 

5-46 

6-56 

7-65 

8-75 

9-84 

10-93 

13-12 

15-31 

17-50 

4 

4-58 

5-72 

6-87 

8-02 

9-16 

10-31 

11-45 

13-75 

16-04 

18-33 

5| 

4-79 

5-98 

7-18 

8-38 

9-58 

10-78 

11-97 

14-37 

16-77 

19-16 

6  in. 

5-00 

6-26 

7-50 

8-75 

10-00 

11-25 

12-50 

lf>  -00 

17-50 

20-00 

SPECIFIC    GRAVITY. 


401 


TABLE  combining  the  Specific  Gravities  and  other  Properties  of 
Bodies.      Water  the  standard  of  comparison,  or  1000. 


METALS. 

STOKES,  EARTHS,  ETC. 

Names. 

1 

Melting  joints  in  de-  1 
green  of  Fahrenheit. 

iliH 
fifff 

Ultimate  cohesive 
strength  of  an  inch 
Bq.  prism  in  tuns. 

Scale  of  wire-drawing 
ductility. 

\ 

Ratio  of  hardness. 

if 

Ratio  of  power  in  the 
conduction  of  heat. 

Names. 

Specific  gravity. 

i 

.2 

If 

Cubio  feet  in  a  ton. 

;li 

1.5 

I 

Platinum  .   . 

19500 

3280 

;. 

3 

5 

3-8 

Marble,  average 

2730 

170-00 

13 

9-25 

Pure  Gold  .    1  19258 

2016 

, 

1 

1 

1-8 

V 

10-0 

Granite,  ditto    . 

2651 

165-68 

13%  6-2 

Mercury   .   .13500 

. 

Purbeck  stone  . 

2601 

162-56 

13%  9-0 

Lead  ....    1  11352 

612 

•319 

'•81 

V 

Y 

1-0 

6 

1-8 

Portland  ditto   . 

2570 

160-62 

14      4-5 

Pure  Silver  . 

10474 

1873 

2 

2 

2-4 

2 

9-7 

Bristol  ditto  .  . 

2554 

159-62 

14       ... 

Bismuth  .  . 

8923 

476 

•156 

1-45 

2-0 

Millstone 

2484 

155-25 

Copper,  cast  . 
"   wrought 

8788 
8910 

1996 

•193 

8-51 
15-08 

5 

8 

2-8 

V 

8-9 

Paving  stone  .  . 
Craigleith  ditto 

2415 
2362 

150-93 
147-62 

15  * 

5-7 
5-0 

Brass,  cast  . 

7824 

1900 

•210 

8-01 

f  to  any 
(degree 

Grindstone  .  .  . 
Chalk,  Brit.  .   . 

2143 
2781 

133-93 

173-81 

}6% 

6-6 

o-r> 

"      sheet  . 

8396 

12-23 

6 

6 

8-6 

Brick  

2000 

125-00 

17 

Iron,  cast  . 

7264 

2786 

•125 

7'87 

{to  any 
degree 

Coal,  Scotch  .  . 
"    Newcastle 

1300 
1270 

81-15 
79-37 

27^ 
27  Ji 

°:! 

"     bar  .   .    7700 

•137 

25-00 

4 

8 

4-7 

4 

3-7 

"    Staffordsh. 

1240 

77-50 

29 

Steel,  soft  .   |  7833 

.  . 

•133 

58-91 

"    Cannel  .  . 

1238 

77-37 

29* 

"      hard    .1  7816 

ftp  any 

(.degree 

Tin,  cast  .  .    1  7291 

442 

•278 

2-11 

8 

4 

1-2 

5 

3-0 

Zinc,  cast  .   .  )   7190 

773       -329 

5-06 

7 

8 

1-6 

7 

3-6 

TABLE  containing  the  Weight  of  Columns  of  Water,  each  one  foot 
in  length,  and  of  Various  Diameters,  in  Ibs.  avoirdupois. 


Diam      Weight,    j     Diam.     |      Weight 


3  in. 


8  in. 


3-0672 
3-3288 
3.6000 
3-SS20 
4-1748 
4-4784 
4-7923 
5-1180 
5-4540 
5-7996 
6-1572 
6-5214 
6-9024 
7-2912 
7-6903 
8-1012 
8-5212 
8-9532 


9-8484 
10-3126 
10-7356 
11-2704 
11-7660 
12-2712 
12-7831 
13-3152 
13-8540 
14-4024 
14-9616 
15-5316 
16-1124 
16-7028 
17-3052 
17-9172 
18-5412 
19-1748 
19-8192 
20-4744 
21-1404 
21-8172 
22--,ai>,<5 
23-2020 
23-9100 
24-5288 
2.V,r.lM 
26-0988 
268500 


27-6120 

28-3848 
29-1672 


30-7657 
31-6524 
32-4060 
;«-242t 
34-0834 
34-9464 
35-8152 


37T.S2S 
38-4828 
393936 
40-3152 
41-2476 
42-1908 
43-143(5 
44-1084 
45-OS28 
46-OfNO 
47-0640 
48-0703 
49-0884 
50-1168 
51-1543 
52-2048 
53-2644 
54-3348 
55-4760 
56-4804 
57-6108' 
58-7244 
59-8476 
60-9828 
62-1276 
63-2832 
64-4496 
65-6268 
66-8148 
68-0136 


70-4424 
71-6724 
72-9120 
74-1643 

75-4272 


Diam. 
15  in. 


I 

A 


Weight.  |j  Diam.    Weight. 


85-9104 


131-5320 
132-9696 
134-6580 
136-3562 
138-0672 
139-7880 
141-5184 
143-2608 
145-0128 
146-7756 
1485492 


21  in. 


150-2376 
152-1288 
153-9348 
155-7396 
157-5780 
159-4152 
161-2644 
163-1220 


166-8732 
168-7632 
170-6652 
172-5780 
174-5004 
176-4336 
178-3776 
180-3324 
182-2980 
184-2744 
186-2616 
188-2584 
190-2672 
T.'2-2,N-)6 
194-3184 
196-3548 
198-4056 
200-4672 
203-5384 
204-6216 
206-7144 
208-8192 
210-9336 
213-0538 
215-1948 
217-3416 
219-4'JSO 
221-6664 
223-8444 
226-0344 
228-2340 
230-4444 
232-6644 
234-8576 
237-1404 
239-3928 
241-6572 
243-9312 
2-16-2160 


30  in. 


402 


THE   PRACTICAL    MODEL    CALCULATOR. 


TABLE  containing  the  Weight  of  Square  Bar  Iron,  from  1  to  10  feet 
in  length,  and  from  J  of  an  inch  to  6  inches  square. 


11 

LENGTH  OF  THE   BARS  IN  FEET. 

1  foot. 

2  feet. 

3  feet. 

4  feet. 

5  feet. 

6  feet.. 

7  feet. 

8  feet. 

9  feet. 

10  feet. 

MSf 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

| 

0-2 

0-4 

0-6 

0-8 

1-1 

1-3 

1-5 

1-7 

1-9 

2-1 

0-5 

1-0 

1-4 

1-9 

2-4 

2-9 

3-3 

3-8 

4-3 

4-8 

0-8 

1-7 

2-5 

3-4 

4-2 

5-1 

5-9 

6-8 

7-6 

8-5 

1-3 

2-6 

4-0 

5-3 

6-6 

7-9 

9-2 

10-6 

11-0 

13-2 

1-9 

3-8 

5-7 

7-6 

9-5 

11-4 

13-3 

15-2 

17-1 

19-0 

2-6 

5-2 

7-8 

10-4 

12-9 

15-5 

18-1 

20-7 

23-3 

25-9 

1  in. 

3-4 

6-8 

10-1 

13-5 

16-9 

20-3 

23-7 

27-0 

30-4 

33-8 

H 

4-3 

8-6 

12-8 

17-1 

21-4 

25-7 

29-9 

34-2 

38-5 

42-8 

l] 

5-3 

10-6 

15-8 

21-1 

26-4 

31-7 

37-0 

42-2 

47-5 

52-8 

if 

6-4 

12-8 

19-2 

25-6 

32-0 

38-3 

44-7 

51-1 

57-5 

63-9 

1J: 

7-6 

15-2 

22-8 

30-4 

38-0 

45-6 

53-2 

60-8 

68-4 

76-0 

i]i 

8-9 

17-9 

26-8 

35-7 

44-6 

53-6 

62-5 

71-4 

80-3 

89-3 

10-4 

20-7 

31-1 

41-4 

51-8 

62-1 

72-5 

82-8 

93-2 

103-5 

If 

11-9 

23-8 

35-6 

47-5 

59-4 

71-3 

83-2 

95-1 

106-9 

118-8 

2  in. 

13-5 

27-0 

40-6 

54-1 

67-6 

81-1 

94-6 

108-2 

121-7 

135-2 

2A 

15-3 

30-5 

45-8 

61-1 

76-3 

91-6 

106-8 

122-1 

137-4 

152-6 

2i 

17-1 

34-2 

51-3 

68-4 

85-6 

102-7 

119-8 

136-9 

154-0 

171-1 

2} 

19-1 

38-1 

57-2 

76-3 

95-3 

114-4 

133-5 

152-5 

171-6 

190-7 

gf 

21-1 

42-8 

63-4 

84-5 

105-6 

126-7 

147-8 

169-0 

190-1 

211-2 

2| 

23-3 

46-6 

69-9 

93-2 

116-5 

139-8 

163-0 

186-3 

209-6 

232-9 

2£ 

25-6 

51-1 

76-7 

102-2 

127-8 

153-4 

178-9 

204-5 

230-0 

255-6 

2| 

27-9 

55-9 

83-8 

111-8 

139-7 

167-6 

195-7 

223-5 

251-5 

279-4 

3  in. 

30-4 

60-8 

91-2  [121-7 

152-1 

182-5 

212-9 

243-3 

273-7 

304-2 

gi 

33-0 

66-0 

99-0 

132-0 

165-1    198-1 

231-1 

264-1 

297-1 

330-1 

31 

35-7 

71-4 

107-1 

142-8 

178-5   214-2 

249-9 

285-6 

321-3 

357-0 

8| 

38-5 

77-0 

115-5 

154-0 

192-5 

231-0 

269-5   308-0 

346-5 

385-0 

2 

41-4 

82-8 

124-2 

165-6 

207-0 

248-4 

289-8   331-3 

372-7 

414-1 

3§ 

44-4 

88-8 

133-3    177-7 

222-1 

266-5 

310-9 

355-3 

399-8 

444-2 

Bi 

47-5 

95-1 

142-6    190-1 

237-7 

285-2 

332-7 

380-3 

427-8 

475-3 

8— 

50-8 

101-5 

152-3   203-0 

253-8 

304-5 

355-3 

406-0 

456-8 

507-6 

48in. 

54-1 

108-2 

162-3   216-3 

270-4 

324-5 

378-6  432-7 

486-8 

540-8 

4i 

67-5 

115-0 

172-6   230-1 

287-6 

345-1 

402-6   460-1 

517-7 

575-2 

4i 

61  -1 

122-1 

183-2   244-2 

305-3 

366-3 

427-4 

488-4 

549-5 

610-6 

64-7 

129-4 

194-1    258-8 

323-5 

388-2 

452-9 

517-6 

582-3 

647-0 

4I 

68-4 

136-9 

205-3   273-8 

342-2 

410-7 

479-1 

547-6 

616-0 

684-5 

4£ 

72-3 

144-6 

216-9   289-2 

361-5   433-8 

506-1 

578-4 

650-7 

723-1 

4f 

76-3 

152-5 

228-8    305-1 

381-3 

457-6 

533-8 

610-1 

686-4 

762-6 

4 

80-3 

160-7 

241-0   321-3 

401-7 

482-0 

562-3 

642-7 

723-0 

803-3 

6  in. 

84-5 

169-0 

253  -4  t  337-9 

422-4 

506-9 

591-4 

675-8 

760-3 

844-8 

5* 

93-2 

186-3 

279-5  i  372-7 

465-8 

559-0 

652-2 

745-3 

838-5 

931-7 

&1 

102-2 

204-5 

306-7  i  409-0 

511-2 

613-4 

715-7 

817-9 

920-2 

1022-4 

5| 

111-8 

223-5 

335-3  !  447-0 

558-8 

670-5 

782-3 

894-0 

1005-8 

1117-6 

6  in. 

121-7 

243-3 

365-0   486-7 

608-3 

730-0 

841-6   973-3 

1009-5 

1216-6 

TABLE  of  the  Weight  of  a  Square  Foot  of  Sheet  Iron  in  Ibs.  avoirdu- 
pois, the  thickness  being  the  number  on  the  wire-gauge.     No.  1 
of  an  inch;  No.  4,  \;  No.  11,  J,  $c. 


s 


No.  on  wire-gauge 

1 

2 

3 

4 

5 

6|    7 

8 

9 

10 

11 

Pounds  avoir  j  12-5 

12  |11 

10 

9 

8  |  7-5 

7 

6 

5-68 

5 

No.  on  wire-gauge 

12 

13J14 

15 

16 

17 

18 

19 

20 

21 

22 

Pounds  avoir  

4-62 

4-31 

4 

3-95 

3 

2-5 

2-18    l-93jl-62|  1-5 

1-37 

SPECIFIC    GRAVITY. 


403 


TABLE  of  the  Weight  of  a  Square  Foot  of  Boiler  Plate  Iron,  from 
J  to  1  inch  thick,  in  Ibs.  avoirdupois. 


i  & 

* 

T5S 

1 

iV 

* 

& 

l|  « 

1 

if 

I 

if 

lin. 

5  7-5 

10 

12-5 

15 

17-6 

20 

22-5 

25  27-5 

30 

32-5 

35 

37-5 

40 

TABLE  containing  the  Weight  of  Hound  Bar  Iron,  from  1  to  10  feet 
in  length,  and  from  J  of  an  inch  to  6  inches  diameter* 


Inches 
diameter. 

LENGTH   OF  THE   BARS   IN   FEET. 

1  foot. 

2  feet. 

3  feet. 

4  feet. 

5  feet. 

6  feet. 

7  feet. 

8  feet. 

9  feet. 

10  feet. 

Lbs. 

Lbs; 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

0-2 

0-3 

0-5 

0-7 

0-8 

1-0 

1-2 

1-3 

1-5 

1-7 

0-4 

0-7 

1-1 

1-5 

1-9 

2-2 

2-6 

3-0 

3-4 

3-7 

j 

0-7 

1-3 

2-0 

2-7 

3-3 

4-0 

4-6 

5-3 

6-0 

6-6 

1-0 

2-1 

3-1 

4-2 

5-2 

6-3 

7-3 

8-3 

9-4 

10-4 

\ 

1-5 

3-0 

4-5 

6-0 

7-5 

9-0 

10-5 

11-9 

13-4 

14-9 

i 

2-0 

4-1 

6-1 

8-1 

10-2 

12-2 

14-2 

16-3 

18-3 

20-3 

1  in. 

2-7 

5-3 

8-0 

10-6 

13-3 

15-9 

18-6 

21-2 

23-9 

26-5 

H 

3-4 

6-7 

10-1 

13-4 

16-8 

20-2 

23-5 

26-9 

30-2 

33-6 

11 

4-2 

8-3 

12-5 

16-7 

20-9 

25-0 

29-2 

33-4 

37-5 

41-7 

If 

5-0 

10-0 

15-1 

20-1 

25-1 

30-1 

35-1 

40-2 

45-2 

50-2 

If 

6-0 

11-9 

17-9 

23-9 

29-9 

35-8 

41-8 

47-8 

53-7 

59-7 

If 

7-0 

14-0 

21-0 

28-0 

35-1 

42-1 

49-1 

56-1 

63-1 

70-1 

If 

8-1 

16-3 

24-4 

32-5 

40-6 

48-8 

56-9 

65-0 

73-2i      81-3 

l| 

9-3 

18-7 

28-0 

37-3 

46-7 

56-0 

65-3 

74-7 

84-0  !      93-3 

2  in. 

10-6 

21-2 

31-8 

42-5 

53-1 

63-7 

74-3 

84-9 

95-5     106-2 

2* 

12-0 

24-0 

36-0 

48-0 

59-9 

71-9 

83-9 

95-9 

107-9     119-9 

2 

13-4 

26-9 

40-3 

53-8 

67-2 

80-6 

94-1 

107:5 

121-0 

134-4 

2 

15-0 

30-0 

44-9 

60-0 

74-9 

89-9 

104-8 

119-8 

134-8 

149-8 

2 

• 

16-7 

33-4 

50-1 

66-8 

83-5 

100-1 

116-8 

133-6 

150-2 

166-9 

2 

• 

18-3 

36-6 

54-9 

73-2 

91-5 

109-8 

128-1 

146-3 

164-6 

182-9 

2 

• 

20-1 

40-2 

60-2 

80-3 

100-4 

120-5 

140-5 

160-6 

180-7 

200-8 

2 

- 

21-9 

43-9 

65-8 

87-8 

109-7 

131-7 

153-6 

175-6 

197-5 

219-4 

3  in. 

23-9 

47-8 

71-7 

95-6 

119-4!  143-3 

167-2 

191-1 

215-0 

238-9 

3 

25-9 

51-9 

77-8 

103-7 

129-6    155-6 

181-5 

207-4 

233-3 

259-3 

3 

28-0 

56-1 

84-1 

112-2 

140-2'  168-2 

196-3 

224-3 

253-4 

280-4 

3 

• 

30-2 

60-5 

90-7 

121-0 

151-2!  181-4 

211-7 

241-9 

272-2 

302-4 

3 

32-5 

65-0 

97-5 

130-0 

162-6 

195-1 

227-6 

260-1 

292-61   325-1 

3 

• 

34-9 

69-8 

104-7 

139-5 

174-4 

209-3 

244-2 

279-1 

314-0 

348-9 

3 

• 

37-3 

74-7 

112-0 

149-3  i  186-7 

224-0 

261-3 

298-7 

336-0 

373-3 

3 

" 

39-9 

79-7 

119-6 

159-5 

199-3 

239-2 

279-0 

318-9 

358-8 

398-6 

4  in. 

42-5 

84-9  i  127-4 

169-9 

212-3 

254-8 

297-2 

339-7 

382-2 

424-6 

4£ 

45-2 

90-3  ;  135-5 

180-7 

225-9 

271-0 

316-2 

361-4 

406-6 

451-7 

H 

48-0 

95-9 

143-9 

191-8 

239-8 

287-7 

335-7 

383-6 

431-6 

479-5 

i 

50-8 

101-6 

152-4 

203-3 

254-1 

304-9 

355-7 

406-5 

457-3 

5082 

53-8 

107-5 

161-3 

215-0 

268-8 

322-6 

376-3 

430-1 

483-8 

537-6 

4|^ 

56-8 

113-6 

170-4 

227-2 

283-9 

340-7 

397-5 

454-3 

511-1 

567-9 

4f 

60-0 

119-8 

179-7 

239-6 

299-5 

359-4 

419-3 

479-2 

539-1 

599-0 

4J 

63-1 

126-2 

189-3 

252-4 

315-5 

378-6 

441-7 

504-8 

567-8 

630-9 

5  in. 

66-8 

1,33-5 

200-3 

267-0 

333-8 

400-5 

467-3 

534-0 

600-8 

667-5 

ft 

73-2 

146-3 

219-5 

292-7 

365-9 

439-0 

512-2 

585-4 

658-5 

731-7 

6| 

80-3 

160-6 

240-9 

321-2 

401-5 

481-8 

562-1 

642-4 

722-7 

803-0 

5f 

87-8 

175-6 

263-3 

351-1 

438-9 

526-7 

614-4 

702-2 

790-0 

877-8 

6  in. 

95-6 

191-1 

286-7 

382-2 

477-8 

573-3 

668-9 

764-4 

860-0 

955-5 

TABLE  of  the  Weight  of  Oast  Iron  Plates,  per  Superficial  Foot,fi 
one-eighth  of  an  inch  to  one  inch  thick. 


rom 


X  inch. 

%  inch. 

%  inch. 

Yz  inch. 

%  inch. 

%  inch. 

Ibs.  oz. 
29   0 

%  inch. 

1  inch. 

Ibs.   01. 

4  13| 

Ibs.  oz. 

9  lOf 

Ibs.    oz. 
14   8 

Ibs.    oz. 
19    5| 

Ibs.  oz. 

24  2| 

Ibs.  oz. 

33  13| 

Ibs.  oz. 

38  lOf 

434 


THE    PRACTICAL   MODEL   CALCULATOR. 


TABLE  containing  the  Weight  of  Cast  Iron  Pipes,  1  foot  in  length. 


"S 

fe    fl    M 

THICKNESS 

IN  INCHES. 

1|1 

1 

* 

f 

f 

1 

1  inch. 

u 

H 

3 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

It 

6-9 

9-9 

fl1 

8-8 

12-3 

16-1 

20-3 

2t 

10-6 

14-7 

19-2 

23-9 

3" 
? 
? 

? 
g 

* 
$ 

# 

10A 

124 
14-2 
16-8 
18-0 
19-8 
21-6 
23-5 
25-3 
27-2 
29-0 
30-8 
32-9 
34-5 
36-3 
38-2 

17-2 
19-6 
22-1 
24-5 
27-0 
29-5 
31-9 
344 
36-8 
39-1 
41-7 
444 
46-6 
49-1 
51-5 
54-0 

22-2 
25-3 
284 
314 
34-5 
37-6 
40-7 
43-7 
46-8 
49-9 
52-9 
56-2 
59-1 
62-1 
65-2 
68-2 

27-6 
31-3 
35-0 
38-7 
42-3 
46-0 
49-7 
534 
56-8 
60-7 
644 
68-3 
71-8 
75-5 
79-2 
82-8 

33-3 
37-6 
41-9 
46-2 
50-5 
54-8 
59-1 
634 
67-7 
72-0 
76-2 
80-8 
84-8 
89-1 
934 
97-7 

39-3 
44-2 
49-1 
54-0 
58-9 
63-8 
68-7 
734 
78-5 
83-5 
884 
93-5 
98-2 
103-1 
108-0 
112-9 

45-6 
51-1 
56-6 
62-1 
67-6 
73-2 
78-7 
84-2 
89-7 
95-3 
100-8 
106-5 
111-8 
1174 
122-8 
128-4 

644 
70-6 
76-7 
82-8 
88-8 
95-1 
101-2 
1074 
113-5 
119-9 
125-8 
131-9 
138-1 
144-2 

II2 

564 

71-3 

86-5 

102-0 

117-8 

133-9 

150-3 

Uj 

12 



58-9 
61-3 

74-3 
77-4 

90-1 
93-6 

106-3 
110-6 

122-7 
127-6 

1394 
145-0 

1564 

162-6 

13 

82-7 

101-2 

118-2 

137-4 

154-1 

173-5 

14 

89-5 

108-2 

126-5 

146-2 

165-3 

185-2 

15 

95-2 

115-7 

135-3 

156-2 

176-2 

198-1 

16 

123-3 

143-1 

166-1 

187-5 

211-3 

17 

130-2 

152-5* 

178-5 

198-2 

293.4 

18 

137-0 

161-2 

185-3 

209-1 

235-6 

19 

169-2 

195-7 

222-3 

247-1 

20 

178-1 

205-2 

233-2 

259-0 

21 

214-1 

243-5 

273-2 

22 

223-0 

254-8 

285-4 

23 

2334 

265-5 

298-3 

24 

245-2 

277-5 

310-6 

TABLE  containing  the  Weight  of  Solid  Cylinders  of  Cast  Iron,  one 
foot  in  length,  and  from  |-  of  an  inch  to  14  inches  diameter. 


Diameter  in 
Inches. 

Weight  iu 
Lbs. 

Oiameter  inj    Weight  in   I 
Inches.      !         Lbs. 

Diameter  in 
Inches. 

Weight  in 

Diameter  in 
Inches. 

Weight  iu 
Lbs. 

| 

1-39 

2* 

2048 

4£ 

58-72 

7f 

148-87 

7 

1-88 

3  in. 

22-35 

5  in. 

61-96 

8  in. 

158-63 

1  in. 

247 

gl 

24-20 

51 

64-66 

8£ 

168-15 

1* 

3-13 

4 

26-18 

5| 

68-31 

8J 

179-08 

4 

3-87 

33 

28-23 

5f 

71-00 

8f 

189-00 

if 

4-68 

3£ 

30-36 

5| 

74-98 

9  in. 

200-77 

i* 

5-57 

3f 

32-57 

5| 

78-65 

9J 

211-12 

4 

6-54 

3f 

34-85 

5% 

81-95 

91 

223-70 

if 

7-59 

37. 

37-21 

$ 

85-81 

9f 

235-31 

ii- 

8-71 

4  in. 

39-66 

6  in. 

89-23 

10  in. 

247-87 

2  in. 

9-91 

*l 

41-80 

6i- 

96-82 

101 

273-27 

2A 

11-19 

il 

44-77 

6J 

104-72. 

11  in. 

299-92 

21. 

12-54 

4$ 

47-00 

gi 

112-93 

Hi 

327-81 

2f 

13-98 

4J 

50-19 

7  in. 

12145 

12  in. 

356-93 

2£ 

1549 

4 

52-71 

7i 

130-28 

13 

418-90 

2| 

17-08 

4f             55-92 

7i 

13942 

14 

485-83 

21 

18-74 

SPECIFIC    GRAVITY. 


405 


TABLE  containing  the  Weight  of  a  Square  Foot  of  Copper  and 
Lead,  in  Ibs.  avoirdupois,  from  -fa  to  J-  an  inch  in  thickness,  ad- 
vancing ly  ^. 


Thickness. 

Copper. 

Lead. 

A 

1-45 

1-85 

Jg 

2-90 

3-70 

A 

4-35 

5-54 

i 

5-80 

7-39 

~8       '      A 

7-26 

9-24 

8-71 

11-08 

I  +  A 

10-16 

12-93 

i 

11-61 

14-77 

I  +  A 

13-07 

16-62 

J  +  A            14-52 

18-47 

i  +  A 

15-97 

20-31 

f 

17-41 

22-16 

I  +  A 

18-87 

24-00 

I  +  A 

20-32 

25-85 

!  +  A 

21-77 

27-70 

23-22 

29-55 

TABLE  for  finding  the  Weight  of  Malleable  Iron,  Copper,  and  Lead 
Pipes,  12  inches  long,  of  various  thicknesses,  and  any  diameter 
required. 


Thickness. 

Malleable  Iron. 

Copper. 

Lead. 

^j  of  an  inch. 

•104 

•121 

•1539 

& 

•208 

•2419 

•3078 

fi 

•3108 

•3628 

•4616 

1 

•414 

•4838 

•6155 

1  +  A 

•518 

•6047 

•7694 

I  +  & 

•621 

•7258 

•9232 

\     +    A 

•725 

•8466 

1-0771 

i 

•828 

•9678 

1-231 

RULE. — Multiply  the  circumference  of  the  pipe  in  inches  by  the 
numbers  opposite  the  thickness  required,  and  by  the  length  in  feet ; 
the  product  will  be  the  weight  in  avoirdupois  Ibs.  nearly. 

Required  the  weight  of  a  copper  pipe  12  feet  long,  15  inches  in 
circumference,  J  -f-  ^  of  an  inch  in  thickness. 

•7258  x  15  =  10-817  X  12  =  130-644  Ibs.  nearly. 

TABLE  of  the  Weight  of  a  Square  Foot  of  Millboard  in  Ibs.  avoirdupois 


Thickness  in  inches 

1 

A 

i 

A 

1 

8 

16 

4 

16 

8 

Weight  in  Ibs  

•688 

1-032 

1-376 

1-72 

2-064 

406 


THE  PRACTICAL  MODEL  CALCULATOR. 


TABLE  containing  the  Weight  of  Wrought  Iron  Bars  12  inches  long 
in  Ibs.  avoirdupois. 


Inch. 

Round. 

Square. 

Inch. 

Round. 

Square. 

•163 

•208 

2£ 

16-32 

20-80 

•367 

•467 

24 

18-00 

22-89 

•653 

•830 

2f 

19-76 

25-12 

1-02 

1-30 

21 

21-59 

27-46 

1-47 

1-87 

3 

23-52 

29-92 

2-00 

2-55 

** 

27-60 

35-12 

i 

2-61 

3-32 

32-00 

40-80 

ii 

: 

3-31 

4-21 

8 

36-72 

46-72 

i 

4-08 

5-20 

4 

41-76 

53-12 

i- 

\ 

4-94 

6-28 

41 

47-25 

60-00 

u 

5-88 

7-48 

4£ 

52-93 

67-24 

i 

• 

6-90 

8-78 

4| 

58-92 

74-95 

ii 

8-00 

10-20 

5 

65-28 

83-20 

i. 

; 

" 

9-18 

11-68 

5J 

72-00 

91-56 

2 

10-44 

13-28 

51 

79-04 

100-48 

21 

- 

11-80 

15-00 

i{ 

86-36 

109-82 

2\ 

13-23 

16-81 

6 

94-08 

119-68 

2 

i 

14-73 

18-74 

7 

128-00 

163-20 

TABLE  oft  he  Proportional  Dimensions  ofQ-sided  Nuts  for  Bolts  from 
J  to  2J  inches  diameter. 


Diameter  of  bolts  

i 

3 

; 

5 

I 

? 

1 

I1 

li 

4 

8 

2 

8 

f 

8 

*• 

•^i 

Breadth  of  nuts  

If 

13 

1 

1* 

it 

1JL 

If 

l^f 

21 

16 

lo 

» 

!*> 

" 

Breadth  over  the  angles 

! 

i§ 

li 

1! 

IA 

1H 

^ 

2J 

Ift 

Thickness  

i 

7 

.9 

I 

7 

1 

li 

lj 

17 

is 

16 

1? 

¥ 

^ 

A5 

M 

"> 

Diameter  of  bolts  

is 

11 

I5 

i! 

I7 

2 

21 

21 

5 

U 

AS 

*4 

-*-8 

i 

2 

Breadth  of  nuts  

<U 

fli 

9JJL 

% 

8i 

31 

^1 

4 

16 

•"• 

**!« 

8 

•*n 

Breadth  over  the  angles 

?H 

21 

3| 

3A 

3i 

8f 

4^ 

4f 

Thickness... 

1A 

m 

1M 

2 

2i 

2* 

2i 

24 

TABLE  of  the  Specific  G-ravity  of  Water  at  different  temperatures, 
that  at  62°  being  taken  as  unity. 


70°  F. 

•99913 

52°  F. 

1-00076 

68 

•99936 

50 

1-00087 

66 

•99958 

48 

1-00095 

64 

•99980 

46 

1-00102 

62 

1- 

44 

1-00107 

68 

1-00035 

42 

1-00111 

56 

1-00050 

40 

1-00113 

54 

1-00064 

38 

1-00115 

The  difference  of  temperatures  between  62°  and  39° -2,  where 
water  attains  its  greatest  density,  will  vary  the  bulk  of  a  gallon 
rather  less  than  the  third  of  a  cubic  inch. 


SPECIFIC    GRAVITY. 


407 


TABLE  of  the  Weight  of  Cast  Iron  Balls  in  pounds  avoirdupois, 
from  1  to  12  inches  diameter ',  advancing  by  an  eighth. 


Inches. 

Lbs. 

Inches. 

Lbs. 

Inches. 

Lbs. 

•14 

4| 

14-76 

8£ 

-  84-56 

•20 

4 

15-95 

gl 

88-34 

•27 

5 

17-12 

8f 

92-24 

•37 

W 

18-54 

8j 

96-26 

•47 

6| 

19-93 

9 

100-39 

1- 

•59 

5| 

21-39 

9^ 

104-62 

1} 

•74 

4 

22-91 

9 

108-98 

1| 

•91 

55 

24-51 

9 

113-46 

2 

1-10 

5£ 

26-18 

9; 

118-06 

2^ 

1-32 

57. 

27-91 

9: 

122-77 

2; 

1-57 

6 

29-72 

ii 

127-63 

2- 

1-84 

6* 

31-64 

9* 

132-60 

2, 

2-15 

6f 

33-62 

10 

137-71 

2 

2-49 

6-| 

35-67 

104 

142-91 

2f 

2-86 

6£ 

37-80 

10i 

148-28 

4 

3-27 

6| 

40-10 

lOf 

153-78 

3 

3-72 

6| 

42-35 

io| 

159-40 

3^ 

4-20 

57 

44-74 

lOf 

165-16 

3 

4-72 

7 

47-21 

10| 

171-05 

3 

5-29 

H 

49-79 

10J 

177-10 

3 

5-80 

7| 

52-47 

11 

183-29 

3 

6-56 

7f 

55-23 

11^ 

189-60 

^ 

7-26 

58-06 

11: 

196-10 

« 

p 

8-01 

7f 

60-04 

11 

202-67 

4 

8-81 

71 

64-09 

lit 

209-43 

4^ 

9-67 

» 

67-25 

11 

216-32 

4 

10-57 

8 

70-49 

llf 

223-40 

4 

11-53 

» 

73-85 

11£ 

230-57 

4^ 

12-55 

a 

77-32     12" 

237-94 

4 

13-62 

8| 

80-88   | 

TABLE  of  the  Weight  of  Flat  Bar  Iron,  12  inches  long,  in  Ibs. 


Thickness. 

i 

T3* 

I 

1 

i 

I 

I- 

1 

1  inch. 

£ 

•21 

•31 

•42 

•63 

S. 

•31 

•47 

•63 

•94 

1-26 

1-57 

r 

•42 

•63 

•84 

1-26 

1-68 

2-10 

2-52 

2-94 

!i 

•52 

•78 

1-05 

1-57 

2-10 

2-62 

3-15 

3-67 

4-20 

If 

•57 

•86 

1-18 

1-73 

2-31 

2-88 

3-46 

4-04 

4-62 

• 

l| 

•63 

•94 

1-26 

1-89 

2-52 

3-15 

3-78 

4-41 

5-04 

5 

If 

•73 

•10 

1-47 

2-20 

2-94 

3-67 

4-41 

5-14 

5-87 

iJB 

o 

2 

•84 

•26 

1-68 

2-52 

3-36 

4-20 

5-06 

5-88 

6-72 

.2 

2i 

•96 

•41 

1-89 

2-83 

3-78 

4-72 

5-66 

6-61 

7-56 

d 

2* 

1-05 

•57 

2-10 

3-15 

4-20 

5-25 

6-30 

7-35 

8-40 

5 

H 

1-15 

•73 

2-31 

3-46 

4-62 

5-77 

6-93 

8-08 

9-24 

1 

3 

1-26 

•89 

2-52 

3-78 

5-04 

6-30 

7-56 

8-82 

10-08 

a 

b 

ft 

1-36 

2-04 

2-73 

4-09 

5-46 

6-82 

8-19 

9-55 

10-92 

M 

3? 

1-47 

2-20 

2-94 

4-41 

5-88 

7-35 

8-82 

10-29 

11-76 

3f 

1-57 

2-36 

3-15 

4-72 

6-30 

7-87 

9-45 

11-02 

12-60 

4 

1-68 

2-52 

3-36 

5-04 

6-72 

8-40 

10-08 

11  76 

13-44 

^ 

1-89 

2-83 

3-73 

5-67 

7-56 

9-45 

11-34 

13-23 

15-12 

5 

2-10 

3-15 

4-12 

6-30 

8-40 

10-50 

12-60 

16-70 

17-80 

6 

2-52 

3-78 

5-04 

7-56 

10-08 

12-60 

15-12 

17-64 

20-16 

Weight  of  a  copper  rod  12  inches  long  and  1  inch  diameter  =  3-039  Ibs. 
Weight  of  a  brass  rod  12  inches  long  and  1  inch  diameter  =  2-86  Ibs. 


408 


THE  PRACTICAL  MODEL  CALCULATOR. 


BRASS. —  Weight  of  a  Lineal  Foot  of  Round  and  Square. 


Diameter. 

Weight  of 
round. 

Weight  of 
square. 

Diameter. 

Weight  of 
round. 

Weight  of 
square. 

Inches. 

Lbs. 

Lbs. 

Inches. 

Lbs. 

Lbs. 

| 

•17 

•22 

If 

8-66 

11-03 

•39 

•50 

If 

9-95 

12-66 

•j  • 

•70 

•90 

2 

11-32 

14-41 

| 

1-10 

1-40 

2i 

12-78 

16-27 

| 

1-59 

2-02 

2* 

14-32 

18-24 

i 

2-16 

.  2-75 

2f 

15-96 

20-32 

2-83 

3-60 

2| 

17-68 

22-53 

H 

3-58 

4-56 

2ft 

19-50 

24-83 

ij 

442 

5-63 

2f 

21-40 

27-25 

i| 

5-35 

6-81 

4 

23-39 

29-78 

i| 

6-36 

8-00 

3 

25-47 

32-43 

if 

7-47 

9-51 

STEEL.—  Weight  of  One  Foot  of  Round  Steel. 


Diameter  in 
inches  and 

i 

1 

i 

1 

t 

1 

1 

1* 

H 

If 

U 

U 

If 

If 

2 

parts. 

Weight  in 
Ibs.  and  deci- 

•167 

•376 

•669 

1-04 

1-5 

2-05 

2-67 

3-38 

4-18 

5  -or, 

6-02 

7-07 

8-2 

.Ml 

11-71 

mal  parts. 

TABLES  or  THE  WEIGHTS  or  ROLLED  IRON, 

Per  lineal  foot,  of  various  sections,  illustrated  in  the  accompanying  cuts,  viz. 
Parallel  Angle  Iron,  equal  and  unequal  sides ;  Taper  Angle  Iron ;  Parallel  T 
Iron,  equal  and  unequal  depth  and  width ;  Taper  T  Iron  ;  Sash  Iron  ;  and  Per- 
manent and  Temporary  Rails. 

TABLE  I. — Parallel  Angle  Iron^of  equal  sides.     (Fig.  1.) 


Length  of  sides 
A  B,  in  inches. 

Uniform  thickness 
throughout. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

i  •    ,e*. 

3 

| 

8-0 

2f 

1 

7-0 

4 

3 

5-75 

2! 

5-16ths 

4-5 

2 
If 
li 
1^ 
1: 

ifull 

No.  6  wire-gauge 
8 

3-75 
3-0 
2-5 
1-75 

1-5 

li 

9 

1-25 

1 

10 

1-0 

! 

10 

•875 

a 

11 

•625 

i 

11 

•568 

*         '        12                                 -5 

Fig.  1. 


SPECIFIC   GRAVITY. 


409 


TABLE  II. — Parallel  Angle  Iron,  of  unequal  sides.     (Fig.  2.) 


Length  of  side  A, 
in  inches. 

Length  of  side  B, 
in  inches. 

Uniform 
thickness 
throughout. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Inches. 

8* 

5 

1 

9-75 

3 

5 

8-75 

3 

4 

5-16ths 

7-5 

2-1- 

4 

5-16ths 

6-75 

f 

4 
4 

5-75 
5-5 

2i 

3 

4-75 

*i 

2* 

2 

3-375 

2-875 

1£ 

'  2 

3-16ths 

2-25 

TABLE  III. — Ta^er  Angle  Iron,  of  equal  sides.     (Fig.  3.) 


Length  of  sides, 
A  A,  in  inches. 

Thickness  of 
edges  at  B. 

Thickness  of  root 
atC. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Inches. 

4 

I 

1 

14-0 

3 

| 

&. 

10-375 

2| 

7-16ths 

9-16ths 

8-25 

u 

£ 

£ 

6-5 

2* 

5-16ths,  full 

7-16ths 

5-0 

2 

i-  full 

5-16ths,  full 

3-875 

If 

i 

5-16ths 

3-25 

ll 

1  bare 

5-16ths,  bare 

2-625 

Fig.  3. 


TABLE  IV. — Parallel  |  Iron,  of  unequal  width  and  depth.   (Fig.  4.) 


Width  of 
top  table 
A,  in 
inches. 

Total 
depth 
B,in 
inches. 

Uniform  thick- 
ness of"  top 
table  C. 

Uniform 
thickness  of 
ribD. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Inches. 

Inches. 

5 

6 

% 

^ 

15-75 

4^- 

31 

1 

9.16ths 

13-25 

4 

3 

3. 

3. 

8-875 

ai 

3 

I 

| 

8-25 

3| 

4 

I 

12-5 

2^ 

3 

1 

|. 

7-0 

21 

2 

5-16ths 

ffull 

4-5 

2 

ji 

5-16ths 

5-16ths 

4-0 

If 

2 

i 

J 

3-125 

it 

2 

| 

I 

2-875 

H 

if 

i 

2-375 

1 

3-16ths 

3-16ths 

1-5 

I 

i4 

3-16ths 

3-16ths 

1-125 

Fig.  4. 
A— - 


v... 


-110 


THE  PRACTICAL  MODEL  CALCULATOR. 


TABLE  V. — Parallel  T  Iron,  of  equal  depth  and  width.     (Fig.  5.) 


Width  of  top 
table,  and  total 
depth  A  A. 

Uniform 
thickness 
throughout. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 
i 

5 

7-16ths 

13-75 

4 

.| 

9-75 

3* 

£ 

8-5 

3 

f 

7-5 

21 

5-16ths 

4-625 

2| 

6-16ths 

4-5 

2 

5-16ths 

3-75 

If 

£ 

3-0 

l| 

^.    / 

2-25 

l} 

| 

1-75 

1 

3-16ths 

1-0 

I. 

1 

•725 

f 

i 

•625 

TABLE  VI.— Taper  T  Iron.     (Fig.  6.) 

Fig.  6. 

r-— A-— -3 

ipp^* 


Width  of 
top  table 
A,  in 
inches. 

Total 
depth 
B,  in 
inches. 

Thickness  of 
top  table  at 
root  C. 

Thickness  of 
top  table  at 
edges  IX 

Uniform 
thickness  of 
ribE. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

3 

31 

i 

f 

7-16ths 

8-0 

3 

2f 

7-16ths 

| 

| 

8-0 

j 

3 
2* 

7-16ths 

6-16ths 

5-16ths 

5-25 
6-5 

2 

]i 

ffull 

5-16ths 

.| 

3-5 

2 

4 

5-16ths 

I 

\ 

2-875 

i. 


-E 


TABLE  VII.— Sash  Iron.     (Fig.  7.) 


Total 
depth  A. 

Depth  of 
rebate  B. 

Width  at 
edge  C. 

Greatest 
width  D. 

Weight  of  one 
lineal  foot 
in  Ibs. 

In-lied. 

Inches. 

Inches. 

2 

1 

No.  9  wire-gauge 

5-8ths 

1-75 

14 

f 

7 

9-16ths 

1-625 

U 

a 

6 

9-16ths 

1-25 

li 

1 

10 

9-16ths 

1-125 

1, 

10 

9-16ths 

1-0 

1 

£ 

i 

\ 

•75 

TABLE  VIII. — Rails  equal  top  and  bottom 
Tables.     (Fig.  8.) 


Depth  A,  in 
inches. 

Width  across  top 
and  bottom  BB, 
in  inches. 

Thickness  of 
ribC. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Incites. 

5 

2| 

f 

25-0 

4% 
4 

n 

i 

23-33 
21-66 

U- B 


SPECIFIC   GRAVITY. 


411 


TABLE  IK.— Temporary  Rails.     (Fig.  9.) 


Top  width 
A,  in 
inches. 

Rib  width 
B,  in 
inches. 

Bed  width 
C,  in 

inches. 

Total  depth 
D,  in 
inches. 

Thickness 
of  bed  E. 

Weight  of  one 
lineal  foot 
in  Ibs. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

u 

I 

3 

2 

7-16ths 

9-0 

3 

2£ 

l 

12-0 

if 

I 

4 

3 

^ 

16-0 

2 

* 

4 

3 

I 

17-33 

TABLE  of  Natural  Sines,  Co-sines,  Tangents,  Co-tangents,  Secants, 
and  Co-secants,  to  every  degree  of  the  Quadrant. 


Deg. 

Sines. 

Co-sines. 

Tangents. 

Co-tangents. 

Secants. 

Co-secants. 

Degree. 

0 

•00000 

1-00000 

•00000 

Infinite. 

1-00000 

Infinite. 

90 

1 

•01745 

•99985 

•01746 

57-2900 

1-00015 

57-2987 

89 

2 

•03490 

•99939 

•03492 

28-6363 

1-00061 

28-6537 

88 

3 

•05234 

•99863 

•05241 

19-0811 

1-00137 

19-1073 

87 

4 

•06976 

•99756 

•06993 

14-3007 

1-00244 

14-3356 

86 

5 

•08716 

•99619 

•08749 

11-4301 

1-00382 

11-4737 

85 

6 

•10453 

•99452 

•10510 

9-51236 

1-00551 

9-56677 

84 

7 

•12187 

•99255 

•12278 

8-14435 

1-00751 

8-20551 

83 

8 

•13917 

•99027 

•14054 

7-1153-7 

1-00983 

7-18530 

82 

9 

•15643 

•98769 

•15838 

6-31375 

1-01246 

6-39245 

81 

10 

•17365 

•98481 

•17633 

5-67128 

1-01543 

5-75877 

80 

11 

•19081 

•98163 

•19438 

5-14455 

1-01872 

5-24084 

79 

12 

•20791 

•97815 

•21256 

4-70463 

1-02234 

4-80973 

78 

13 

•22495 

•97437 

•23087 

4-33148 

1-02630 

4-44541 

77 

14 

•24192 

•97030 

•24933 

4-01078 

1-03061 

4-13356 

76 

15 

•25882 

•96593 

•26795 

3-73205 

1-03528 

3-86370 

75 

16 

•27564 

•96126 

.  -28675 

3-48741 

1-04030 

3-62796 

74 

17 

•29237 

•95630 

•30573 

3-27085 

1-04569 

3-42030 

73 

18 

•30902 

•95106 

•32492 

3-07768 

1-05146 

3-23607 

72 

19 

•32557 

•94552 

•34433 

2-90421 

1-05762 

3-07155 

71 

20 

•34202 

•93969 

•36397 

2-74748 

1-06418 

2-92380 

70' 

21 

•35837 

•93358 

•38386 

2-60509 

1-07114 

2-79043 

69 

22 

•37461 

•92718 

•40403 

2-47509 

1-07853 

2-66947 

68 

23 

•39073 

•92050 

•42447 

2-35^85 

1-08636 

2-55930 

67 

24 

•40674 

•91355 

•44523 

2-24004 

1-09464 

2-45859 

66 

25 

•42262 

•90631 

•46631 

2-14451 

1-10338 

2-36620 

65 

26 

•43837 

•89879 

•48773 

2-05030 

1-11260 

2-28117 

64 

27 

•45399 

•89101 

•50952 

1-96261 

1-12233 

2-20869 

63 

28 

•46947 

•88295 

•53171 

1-88073 

1-13257 

2-13005 

62 

29 

•48481 

•87462 

•55431 

1-80405 

1-14335 

2-06266 

61 

30 

•50000 

•86603 

•57735 

1-73205 

1-15470 

2-00000 

60 

31 

•51504 

•85717 

•60086 

1-66428 

1-16663 

1-94160 

59 

32 

•52992 

•84805 

•62487 

1-60033 

1-17918 

1-88708 

58 

33 

•54464 

•83867 

•64941 

1-53986 

1-19236 

1-83608 

57 

34 

•55919 

•82904 

•67451 

1-48256 

1-20622 

1-78829 

56 

35 

•57358 

•81915 

•70021 

1-42815 

1-22077 

1-74345 

55 

36 

•58778 

•80902 

•72654 

1-37638 

1-23607 

1-70130 

54 

37 

•60181 

•79863 

•75355 

1-32704 

1-25214 

1-66164 

53 

38 

•61566 

•78801 

•78129 

1-27994 

1-26902 

1-62427 

52 

39 

•62932 

•77715 

•80978 

1-23490 

1-28676 

1-58902 

51 

40 

•64279 

•76604 

•83910 

1-19175 

1-30541 

1-55572 

50 

41 

•65606 

•75471 

•86929 

1-15037 

1-32511 

1-52425 

49 

42 

•66913 

•74314 

•90040 

1-11061 

1-34561 

1-49448 

48 

43 

•68200 

•73135 

•93251 

1-07237 

1-36706 

1-46628 

47 

44 

•69466 

•71934 

•96569 

1-03553 

1-39012 

1-43956 

46 

45 

•70711 

•70711 

1-00000 

1-00000 

1-41421 

1-41421 

45 

Deg. 

Co-sines. 

Sines. 

Co-tangents. 

Tangents. 

Co-secants. 

Secants. 

Degree. 

412 


MOMENT  OF  INERTIA. 


CORDS,    KNOTS,    NODES,    CHAIN-BRIDGE.  —  ANGULAR   VELOCITY.  —  RADIUS 

OP  GYRATION. 

1.  IF  the  cord  q  NB,  be  fixed  at  the  extremity  B,  and  stretched 
by  a  weight  of  500  Ibs.  at  the  extremity  q,  and  the  middle  knot  or 
node  N,  by  a  force  of  255  Ibs.  pulling  upwards,  under  an  angle 
a  N  b  of  54°  ;  what  is  the  tension  and  position  of  NB. 


Angle?  Nr  =  180°  -  angle  gNP;  and  90°  -«N5  =  6N<?  = 
36°;  cos.  36°  =  -80902. 

^"500*  +  2552  -  2  x  255  x  500  x  cos.  36°  =  329-7  Ibs.,  the 
magnitude  of  the  tension. 

--  =  "891386  =  sine  of  angles  b  N  *,  or  angle  BN  r  = 


63°  2'. 

2.  Between  the  points  A  and  B,  a  cord  10  feet  in  length  is 
stretched  by  a  weight  W  of  500  Ibs.  suspended  to  it  by  a  ring  ; 
the  horizontal  distance  AE  =  6'6  feet,  and  the  vertical  distance 
BE  ==3-2  feet;  required  the  position  of  the"  ring  C,  the  tensions, 
and  directions  of  the  rope.  , 

The  tensions  of  the  cords  AC,  CB  are  equal,  and  angle  AC  b  = 
angle  b  CB. 


MOMENT   OF   INERTIA. 


413 


i 


•JE 


B 


\D 


AD  =  AC  +  CB  =  10  feet. 
-  6-62)  =  7-5126  =  ED;  BD  =  7-5126  -  3-2  =  4-3126 


Dn  =  —<  ^  =  2-1563;    7-5126  :  2-1563  :  :  10  :  | 
2-87  =  CD  =  CB  ;  and  CA  =  10  -  2-87  =  7-13. 


=  coshie  b  CB 


W 


-75132. 

500 


332-7  Ibs., 


the  tension   on   the   cord   CB,   which   is    equal   to   the   tension 
on  AC. 


414 


THE  PRACTICAL  MODEL  CALCULATOR. 


3.  Let  500,000  Ibs.  be  the  whole 
weight  on  a  chain-bridge  whose  span 
AB  =  400  feet,  and  height  of  the  arc 
CD  =  40  feet ;  required  the  tensions 
and  other  circumstances  respecting  the 
chains. 

The  tangent  of  the  angles  of  incli- 
nation of  the  ends  of  the  chain  is 


40  x  2 
2QQ     =  '40000,  the  angle  answer- 

ing to  this  natural  tangent  is  21°  48'. 
The  vertical  tension  at  each  point 
of  suspension  is  =  half  the  weight  = 
250000  ;  the  horizontal  tension  at  the 
points  of  suspension  =  250000  x  cot. 


21°  48' 


625000  Ibs. 


The  whole  tension  at  one  end  will  be 


673146  Ibs. 


v/6250002  +  2500002 

4.  Suppose  the  piston  of  a  steam 
engine,  with  its  rod,  weighs  1000  Ibs. ; 
it  has  no  velocity  at  its  highest  and 
lowest  positions,  but  in  the  middle  the 
velocity  is  a  maximum  and  equal  10  ft. ; 
what  effect  will  it  accumulate  by  virtue 
of  its  inertia  in  the  first  half  of  its 
path,  and  give  out  again  in  the  second 
half ;  and  what  is  the  mean  force  which 
would  be  requisite  to  accelerate  the 
motion  of  the  piston  in  the  first  half 
of  its  path,  which  is  the  same  as  that 
which  it  would  exert  in  the  second  half 
by  its  retardation,  the  length  of  stroke 
being  8  feet. 

According  to  the  principle  of  vis 
viva,  the  effect  which  the  piston  will 
accumulate  by  virtue  of  its  inertia  in 
the  first  half  of  its  path,  and  give 
out  again  in  the  second  half  = 

102 
g  x  32.2  X  1000  =  1552-794  units  of  work.     Half  the  path  of 

the  piston  =  4  feet ;  hence, 

1552*794 

j-^  -  =  388-1985  Ibs.,  the  mean  force. 

MOMENT   OF  INERTIA,  or  the  MOMENT  OF  ROTATION,  or  the 
MOMENT  OF  THE  MASS,  is  the  sum  of  the  products  of  the  particles 


MOMENT   OF   INERTIA. 


415 


of  the  mass  and  the  squares  of  their  distances  from  the  axis  of 
rotation. 

5.  If  a  body  at  rest,  but  capable  of  turning  round  a  fixed  axis  A, 
possesses  a  moment  of  inertia  of  121  units  of  work,  the  measures 
taken  in  feet  and  pounds,  made  to  turn  by  means  of  a  cord  and 
weight  of  86  Ibs.,  lying  over  a  pulley  in  a  path  of  10  feet ;  what 
are  the  circumstances  of  the  motion. 


2  x  36  x  10 

121       -  =  2-439347  feet,  the  angular  velocity  of  the 

body,  which  call  v  ;  so  that  each  point  at  the  distance  of  one  foot 
from  the  axis  of  revolution  will  describe,  after  the  accumulatior 
of  121  units  of  work,  2-44  feet  in  a  second. 

6-2832  =  circumference  of  a  circle  2  feet  in  diameter, 

6-2832 
n.AA     =  2-6  seconds,  the  time  of  one  revolution. 

6.  If  an  angular  velocity  of  3  feet  passes  into  a  velocity  of  7  feet ; 
what  mechanical  effect  will  a  mass  produce  so  moving,  supposing 
the  moment  of  inertia  to  be  200,  the  measures  taken  in  feet  and 
pounds. 

According  to  the  principles  of  vis  viva, 

200 
(72  -  32)  -3-  =  4000  units  of  work,  which  may  be  40  Ibs. 

raised  100  feet,  80  Ibs.  raised  50  feet,  400  Ibs.  raised  10  feet ;  and 
so  on. 

7.  The  weight  of  a 
rotating   mass    B    is 
500  Ibs.,  its  distance 
OB  from  the  axis  of 
rotation    3    feet,  the 
weight  W,  constitut- 
ing the  moving  force, 
90  Ibs.,  its  arm  AO. 
=  00  =  4  feet ;  re- 
quired   the    circum- 
stances of  the  motion 
that  ensues. 

|  90  +  ?  500  1  -5- 

32-2  =  11-53  Ibs., 
the  inert  mass  accele- 
rated by  the  force  of 
W.  And  it  is  well 
known  that  the  force 
divided  by  the  mass 
gives  the  accelera- 
tion. 


416 


THE  PRACTICAL  MODEL  CALCULATOR. 


90 


'"  11-53       7*806,  the  acceleration  of  the  motion  of  W.     The 

7-806 
angular  acceleration  in  a  circle  1  foot  from  the  axis  =  —  -i  —  = 

1-9515. 

After  10  seconds  the  acquired  angular  velocity  will  be 
1-9515  x  10  =  19-515. 

1-9515  x  102 

And  the  corresponding  distance  =  -     —  9  —     -  =  97-575  feet, 

measured  on  a  circle  one  foot  from  0. 

The  space  described  by  the  weight  W  is  -  «  —   ~  =  390-3  feet, 

which  is  the  same  as  the  space  described  by  C.     The  circumfe- 
rence of  a  circle  one  foot  from  C  =  3-1416. 

97-575 

=  31-059  revolutions. 


In  the  rotation  of  a  body  AB  about  a  fixed  axis  0,  all  its  points 
describe  equal  angles  in  equal  times.  -  If  the  body  rotate  in  a  cer- 

6° 

tain  time  through  the  angle  0°,  or  arc  $  =  TQQO*?  radius  =  1; 

and  hence,  *  =  3-141592,  &c. ;  the  elements  of  the  body,  a,  5,  c,  &c., 

at   the   distances 

oa   =   x^  ob   = 

xat  &c.  from  the 

axis,  will  describe 

the  arcs  or  spaces 

a  a,  =  rpx..  bb<  = 


&c.  If  the  angu- 
lar velocity,  that 
is,J;he  velocity  of 
those  points  of 
the  body  which 
are  distant  a  unit 
of  length,  a  foot, 
from  the  axis  of 
revolution,  be  put 
=  z,  then  the  si- 
multaneous velo- 
cities of  the  ele- 
ments of  the  mass  at  the  distances  x^  xa,  xa,  &c.,  will  be, 

Z  Xtf  Z  Xa,  Z  X3,  &C. 

And  if  a  be  the  mass  of  the  element  at  a  ;  b  the  mass  of  the  ele- 
ment at  b  ;  c  the  mass  of  the  element  at  <?,  &c.,  their  vis  viva  will  be, 

(zxy  a,  (zx2)2b,  (zx3)2c,  &c. 
And  the  sum  of  the  vis  viva  of  the  whole  body  = 


MOMENT   OF   INERTIA.  417 

According  to  our  definition,  x* a  -f  x*b  -f  x*c,  &c.  is  the  mo- 
ment of  inertia,  which  may  be  represented  by  R ;  then  z3  R  is  the 
vis  viva  of  a  body  revolving  with  the  angular  velocity  z.  There- 
fore, to  communicate  to  a  body  in  a  state  of  rest  an  angular  velo- 
city 2,  a  mechanical  effect  F  s,  or  force  X  space  =  J  the  vis  viva, 
must  be  expended ;  that  is,  F  s  =  J  z2  R,  or,  which  is  the  same  thing, 
a  body  performing  the  units  of  work  F  s,  passes  from  the  angular 
velocity  z  to  a  state  of  rest.  In  general,  if  the  initial  angular 
velocity  =  v9  and  the  terminal  angular  velocity  =  2,  the  units  of 
work  will  be, 

g2  —  v2 

.E  8  ==s          o          X  •">• 

7%e  moment  of  inertia  of  a  body  about  an  axis  not  passing 
through  the  centre  of  gravity  is  equivalent  to  its  moment  of  inertia 
about  an  axis  running  parallel  to  it  through  the  centre  of  gravity, 
increased  by  the  product  of  the  mass  of  the  body  and  the  square  of 
the  distance  of  the  two  centres. 

It  is  necessary  to  know  the  moments  of  inertia  of  the  principal 
geometrical  bodies,  because  they  very  often  come  into  application 
in  mechanical  investigations.  If  these  bodies  be  homogeneous,  as 
in  the  following  we  will  always  suppose  to  be  the  case,  the  particles 
of  the  mass  M,,  M2,  &c.  are  proportional  to  the  corresponding  par- 
ticles of  the  volume  V±,  V3,  &c. ;  and  hence  the  measure  of  the 
moment  of  inertia  may  be  replaced  by  the  sum  of  the  particles  of 
the  volume,  and  the  squares  of  their  distances  from  the  axis  of 
revolution.  In  this  sense,  the  moments  of  inertia  of  lines  and  sur- 
faces may  also  be  found. 

If  the  whole  mass  of  a  body  be  supposed  to  be  collected  into  one 
point,  its  distance  from  the  axis  may  be  determined  on  the  suppo- 
sition that  the  mass  so  concentrated  possesses  the  same  moment 
of  inertia  as  if  distributed  over  its  space.  This  distance  is  called 
the  radius  of  gyration,  or  of  inertia.  If  R  be  the  moment  of  iner- 
tia, M  the  mass,  and  r  the  radius  of  gyration,  we  then  have  M  r2  — 

np 

R,  and  hence  r  —  \  TT-     We  must  bear  in  mind  that  this  radius  by 

no  means  gives  a  determinate  point,  but  a  circle  only,  within  whose 
circumference  the  mass  may  be  considered  as  arbitrarily  distributed. 

If  into  the  formula  Rj  =  R  +  M  e2,  expressed  in  the  words  above 
printed  in  italics,  we  introduce  R  =  M  r2  and  Rj  =  M  rt2,  we 
obtain  r*  =  r2  4-  e2 ;  that  is,  the  square  of  the  radius  of  gyration 
referred  to  a  given  axis  =  the  square  of  the  radius  of  gyration 
referred  to  a  parallel  line  of  gravity,  plus  the  square  of  the  dis- 
tance between  the  two  axes. 

Wheel  and  axle. — The  theory  of  the  moment  of  inertia  finds  its 
most  frequent  application  in  machines  and  instruments,  because  in 
these  rotary  motions  about  a  fixed  axis  are  those  which  generally 
present  themselves. 

27 


418 


THE  PRACTICAL  MODEL  CALCULATOR. 


If  two  weights,  P  and  Q,  act  on  a  wheel  and  axle  ACDB,  with 
the  arms  CA  =  a  and  DB  =  b  through  the  medium  of  perfectly 
flexible  strings,  and  if  the  radius  of  the  gudgeons  be  so  small  that 
their  friction  may  be  neglected,  it  will  remain  in  equilibrium  if  the 
statical  moments  P  .  CA  and  Q  .  DB  are  equal,  and  therefore 
P  a  =•  Q  b.  But  if  the  moment  of  the  weight  P  is  greater  than 
that  of  Q,  therefore  P  a  >  Q  5,  P  will  descend  and  Q  ascend ;  if 
P  a  <  Q  b,  P  will  ascend  and  Q  descend.  Let  us  now  examine  the 


O 


conditions  of  motion  in  the  case  that  P  a  >  Q  b.  The  force  corre- 
sponding to  the  weight  Q  and  acting  at  the  arm  b  generates  at  the 

arm  a  a  force  — ,  which  acts  opposite  to  the  force  corresponding 
to  the  weight  P,  and  hence  there  is  a  residuary  moving  force 
P  —  —  acting  at  A.  The  mass  —  is  reduced  by  its  transference 

Q  b2 
from  the  distance  b  to  that  of  a  to  — -^  '•>  hence  the  mass  moved  by 

p  —  5L-  is  M  —  (p  _| -\  -r-  g^  or,  if  the  moment  of  inertia  of 


MOMENT   OF   INERTIA.  419 


the  wheel  and  axle  without  the  weights  P  and  Q  =  -  ,  and,  there- 

G#2 
fore,  its  inert  mass  reduced  to  A  =  —  j,  we  have,  more  exactly, 


M  =  (P  +      r  +          }  +  g  =  (Pa*  +  Qi*  + 

From  thence  it  follows  that  the  accelerated  motion  of  the  weight 
P,  together  with  that  of  the  circumference  of  the  wheel,  namely, 

P_^ 

moving  force  __  a  2  Pa-—  Q  6 


2 
mass  C 


on  the  other  hand,  the  accelerated  motion  of  the  ascending  weight 
Q,  or  of  the  circumference  of  the  axle,  is, 
b  Pa-  Q6 

*  =  a?  ^  Pa»+QP+Gy»^- 

The  tension  of  the  string  by  P  is  S  =  P  --  -  =  P  (l  -  -), 

y  y 

that  of  the  string  by  Q  is  T  =  Q  H  --  -  =  Q  (l  +  -)  ;  hence  the 

9  9 

pressure  on  the  gudgeon  is, 

Q    ,   T-P  +  0       P^4.Ql-P  +  Q  (P 

y  "T"  ~Q~Pa2 

the  pressure,  therefore,  on  the  gudgeons  for  a  revolving  wheel  and 
axle  is  less  than  for  one  in  a  state  of  equilibrium.  Lastly,  from 
the  accelerating  forces  p  and  ^,  the  rest  of  the  relations  of  motion 
may  be  found  ;  after  t  seconds,  the  velocity  of  P  is  v  —  p  t,  of  Q 
is  v±  =  q  t,  and  the  space  described  by  P  is  s  =  \  p  t2,  by  Q  is 


Let  the  weight  P  at  the  wheel  be  =  60  Ibs.,  that  at  the  axle 
Q  =  160  Ibs.,  the  arm  of  the  first  CA  =  a  =  20  inches,  that  of 
the  second  DB  =  6  =  6  inches  ;  further,  let  the  axle  consist  of  a 
solid  cylinder  of  10  Ibs.  weight,  and  the  wheel  of  two  iron  rings 
and  four  arms,  the  rings  of  40  and  12  Ibs.,  the  arms  together  of 
15  Ibs.  weight;  lastly,  let  the  radii  of  the  greater  ring  AE  = 
20  and  19  inches,  that  of  the  less  FG  =  8  and  6  inches  ;  required 
the  conditions  of  motion  of  this  machine.  The  moving  force  at 
the  circumference  of  the  wheel  is, 

p  -  -  Q  =  60  -  J.  160  =  60  -  48  =  12  Ibs., 

(t  a\J 

the  moment  of  inertia  of  the  machine,  neglecting  the  masses  of  the 
gudgeons  and  the  strings,  is  equivalent  to  the  moment  of  inertia 

W  b2      10  .  62 
of  the  axle  =  —5-  =  ~o~~  —  180,  plus  the  moment  of  the  smaller 

ring  =  RI  ^29+  ^  =  IK?!^?2)  ^  600>  plus  the  moment  of 


420  THE   PRACTICAL  MODEL   CALCULATOR. 

40  (202  +  192) 
the  larger  ring  =  -       — ^—  =  15220,  plus  the  moment  of 

, ,                         .A  (P  3  —  p 3)      A  p 2  -f-  p,  pa  4-  P  2) 
the  arms,  approximately  =    Q  /     __ — rr  =  — — pM —  = 

15  (192  +  19  x  8  +  82) 

— 3—  -  =  2885 ;  hence,  collectively,  G  f  =  180  + 

600  +  15220  +  2885  =  18885,  or  for  foot  measure  = 

131-14.  The  collective  mass,  reduced  to  the  circumference  of  the 
wheel  is, 

[60  +  160  (|)2  +—><,= 

(60  +  160  x  0-09  +  ^^)  0-031  =  121-61  x  0-031  =  337  Ibs. 

Accordingly,  the  accelerated  motion  of  the  weight  P,  together 
with  that  of  the  circumference  of  the  wheel,  is, 
b 

"'19  12 

P  ''        p  .j.  Q  52  .j.  Q. y2  9    —  g^y    =  3-183  feet;  on  the  other 

b  6 

hand,  that  of  Q  is  q  =  -p  =  ^  3-183  =  0-954  feet;  further,  the 

tension  of  the  string  by  P  is  =  (l  -  -}  P  =  (l  -  -g^")   60   = 

54-07  Ibs. ;  that  by  Q,  on  the  other  hand,  Q  =   (l  -f  -)  Q  = 

(1  +  0-925  X  0-032)  160  =  1-030  160  =  164-8  Ibs. ;  and  con- 
sequently the  pressure  on  the  gudgeons  S  +  T  =  54-06  -f  164-80  = 
218-86  Ibs.,  or  inclusive  of  the  weight  of  the  machine  =  218-86  + 
77  =  295-86  Ibs.  After  10  seconds,  P  has  acquired  the  velocity 
p  t  =  3-084  X  10  =  30-84  feet,  and  described  the  space  #  = 

—  =  30-84  X  5  =  154-2  feet,  and  Q  has  ascended  a  height  -  s  = 

0-3  X  154-2  =  46-26  feet. 

The  weight  P  which  communicates  to  the  weight  Q  the  accele- 
rated motion  q  —  p  aa  -j.  Q  y  ^_  Q.  yg  9->  ma7  also  ^e  replaced  by 

another  weight  P4,  without  changing  the  acceleration  of  the  motion 
Q,  if  it  act  at  the  arm  a^  for  which, 

P,^  -  Q6          =         P^~  Qb 

Va2  +  Qb2  -f  Gv2 
The  magnitude  -      pa  _  Q^ >  represented  by  k,  and  we  ob- 


tain  at2,  —  ka±  —  —  -         — p —         — ,  and  the  arm  m  question, 


MOMENT   OF  INERTIA.  421 


We  may  also  find  by  help  of  the  differential  calculus,  that  the  mo- 
tion of  Q  is  most  accelerated  by  the  weight  P,  when  the  arm  of  the 
latter  corresponds  to  the  equation  P  a2  —  2  Q  a  b  =  Q  b2  -f  G  y2, 
therefore, 


p   r  wv  p 

The  formula  found  above  assumes  a  complicated  form  if  the  fric- 
tion of  the  gudgeons  and  the  rigidity  of  the  cord  are  taken  into 
account.  If  we  represent  the  statical  moments  of  both  resistances 

by  F  r,  we  must  then  substitute  for  the  moving  force  P Q,  the 

value  P ,  whence  the  acceleration  of  Q  comes  out, 


Q6+Fr 


The  weights  P  =  30  Ibs.  Q  =  80  Ibs.  act  at  the  arms  a  =  2  feet, 
and  b  =  J  foot  of  a  wheel  and  axle,  and  their  moments  of  inertia 
G#2  amount  to  60  Ibs.  ;  then  the  accelerated  motion  of  the  ascend- 
ing weight  Q  is, 

30x2x|-80x  (|)2  30  -  20  322 

?  ~  30  x  22  +  80x(£)2+  60  g  ''=  120  +  20  -f  60  S2'2      ''   200  ~ 
1-61  feet.     But  if  a  weight  P±  =  45  Ibs.  generates  the  same  acce- 
leration in  the  motion  of  Q,  the  arm  of  Pt  is  then, 

k         /TV       80  x  J  (i  +  A:)  +  60  200 

fli  =  2  ±  <J\2)  45  ~'  or  as  *  ==  60-40  = 

10,  at  is  =  5  db  J25  -  y  =  5  =b  J-  11-358  =  5  ±  3-786  =  8-786 

feet,  or  1-214  feet. 

The  accelerated  motion  of  Q  comes  out  greatest  if  the  arm  of 
the  force  or  radius  of  the  wheel  amount  to, 

_|x80        /740T2      2QT"60      4        116      24      4  +  x/40 
a~"    30     "'"N/VSO/  30""~3  +  N/9"9=  3 

/30  x  1-7207  -  20x  31-621 

3-4415  feet,  and  q  is  =  (30-XT3-4415)2  +  8o)  9  =  435^32  9   = 
2-339  feet. 

The  statical  moment  of  the  friction,  together  with  the  rigidity 
of  the  string,  is  F  r  =  8  ;  then,  instead  of  Q  b,  we  must  put  Q  b  -f 
Fr  =  40  +  8  =  48  ;  whence  it  follows  that, 


a  =        -f  )    4-      =  l'6  +  ^/fraa?  =  3-886,  and  the  cor- 

respondent maximum  accelerating  force 

30  x  1-943  -  8  x  J  -  20  34-29      oo  o 

*  =  -    30x(3-886)2+80    "  9  =     533~  X  32'2  =  2*071  feet 


•422 


WEIGHT,  ACCELERATION,  AND  MASS. 

PARALLELOGRAM  OF  FORCES. — THE  PRINCIPLE  OF  VIRTUAL  VELOCITIES. 
— MECHANICAL  POWERS :  CONTINUOUS  CIRCULAR  MOTION,  GEARING, 
TEETH  OF  WHEELS,  DRUMS,  PULLEYS,  PUMPING  ENGINES,  ETC. 

1.  IP  a  weight  of  10  Ibs.,  moved  by  the  hand,  ascends  with  a  3 
feet  acceleration,  what  is  the  pressure  on  the  hand  ? 

10  (1  +  gj^)  =  10-93168  Ibs. 

If  a  weight  of  10  Ibs.,  moved  by  the  hand,  descends  with  a  3  feet 
acceleration,  the  pressure  on  the  hand  will  be  9-06832  Ibs.,  for  then 

10  (1  -  3232)  =  9-06832. 

If  w  be  the  weight  of  the  mass  acted  upon  by  the  force  of  the 
hand,  and  also  by  the  force  of  gravity,  as  g  —  32-2,  the  mass 

w 
moved  by  the  sum  or  difference  of  these  forces  will  be  =  - .  If  P 

9 
be  the  pressure  on  the  hand,  and  p  its  acceleration,  the  body  falls 

w 

with  the  force  —  p ;  it  also  falls  with  the  force  iv  —  P ;  hence, 
9 

w--p  =  -p     ...  p  =  (!-•£)  w. 

9*  9} 

When  the  body  is  ascending,  then  p  is  negative, 

w  f> 

and  w  +  P  =  -  (-  p)     .:  P  =  (1  +  J)  w. 

2.  If  a  body  of  200  Ibs.  be  moved  on  a  smooth  horizontal  track, 
by  the  joint  action  of  two  forces,  and  describes  a  space  of  10  feet 
in  the  first  second,  what  is  the  amount  of  each  of  these  forces  ;  the 
first  makes  an  angle  of  35°  with  the  track  upon  which  the  body 
moves,  and  the  other  an  angle  of  50°  ? 

In  solving  this  question,  the  natural  sines  of  the  angles  35°,  50°, 
and  of  their  sum  85°,  will  be  required.  We  shall  first  take  these 
from  the  table : 

sin.  35°  =  -57358 

sin.  50°  =  -76604 

sin.  85°  =  -99619. 

The  acceleration  is  =  20  feet,  that  is,  twice  the  space  passed  over 
in  the  first  second, 

— 2  =s  the  masSj  and  Q^.O  ,x  20  =  124-224  Ibs.,  the  force  of 

the  resultant,  in  the  direction  of  the  track  upon  which  the  body 
moves. 


WEIGHT,    ACCELERATION,    AND    MASS. 


423 


One  of  the  components 
The  other  component  ; 


124-224  sin.  35° 
:  sin7(350  +  50°) 
124-224  sin.  50° 


71-52  Ibs. 


95-52  Ibs. 


sin.  (35°  +  50°) 

These,  and  the  like  results,  may  be  obtained  with  greater  ease 
by  logarithms. 


Log.  124-224 
Log.  sin.  35° 

Log.  sin.  85° 
Log.  of  71-52413 

Log.  124-224 
Log.  sin.  50 


o 


Log.  sin.  (85°) 
Log.  of  95-5247 


2-0942055 
9-7585913 

11-8527968 
9-9983442 

1-8544526 

2-0942055 

9-8842540 

11-9784595 
9-9983442 


1-9801153 

3.  A  carriage  weighing  8000  Ibs.  is  moved  forward  by  a  force  /t 
of  500  Ibs.  upon  a  horizontal  surface  AB ;  during  the  motion,  two 
resistances  have  to  be  overcome,  one  horizontal  of  100  Ibs.,  the 
amount  of  friction,  represented  in  the  figure  by/8,  the  other  /3  of 

. 


200  Ibs.  acting  downwards  ;  the  angles /3  nfa  and/  n  m,  which  the 
directions  of  these  forces  make  with  the  horizon,  are  61°  and  21° 
respectively :  it  is  required  to  know  what  work  the  force  /±  will 
perform  by  converting  a  5  feet  initial  velocity  of  the  carriage  into 
a  20  feet  velocity. 

If  we  put  x  =  n  m,  the  distance  the  carriage  moves  in  passing 
from  a  5  to  a  20  feet  velocity, 

The  work  of  the  force  /  =/t  X  nq  =  500  X  cos.  21°  x  x. 

The  work  of  the  force/,  =  (— /8)  X  nm  =  —  100  X  x. 

The  work  of  the  force/;  =  (-/a)  X  np  =  —  200  x  cos.  61°  x  x. 


424 


THE  PRACTICAL  MODEL  CALCULATOR. 


Consequently,  the  work  of  the  effective  force  will  be  269-828  X 
x  =   {500  X  -94358  -  100  —  200  X  -48481}  x,  since  the  natu- 
ral'cosine  of  21°  =  -93358,  and  the  natural  cosine  of  61°  =  -48481. 
But  according  to  the  principle  of  via  viva,  the  work  done  is 
equal  to 

202  —  52 

x  8000  =  46589-82. 


64-4 
.-.  269-828  x  x  =  46589-82  and  x 


46589-82 


269-828   " 
772-665  feet,  the  space  passed  over  by  the  carriage. 

This  question  is  solved  on  the  PRINCIPLE  OF  VIRTUAL  VELOCITIES, 
which  we  shall  explain,  as  it  is  of  essential  service  in  practical 
mechanics. 

This  explanation  depends  on  what  is  technically  termed  the 
"  Parallelogram  of  Forces." 


When  a  material  point  0,  is  acted  upon  by  two  forces  f^f^  whose 
directions  0/t,  0/3,  make  with  each  other  an  angle,  if  0/a,  0/3  re- 
present the  magnitudes  and  directions  of  the  forces,  the  diagonal 
of  the  parallelogram  0  ft  f3  fz  represents  the  resultant  in  magnitude 
and  direction  ;  that  is,  the  diagonal  represents  a  single  force  equal 
to  the  combined  actions  of  the  forces  represented  by  the  sides. 
And  if  the  sides  of  the  parallelogram  represent  the  accelerations 
of  the  forces,  the  diagonal  represents  the  resultant  acceleration. 
Draw  through  0,  two  axes  OX  and  OY,  at  right  angles  to  each 
other,  and  resolve  the  forces  ft  and  /2,  as  well  as  their  resultant  /„, 
into  components  in  the  directions  of  these  axes  ;  namely,  /4  into  n^ 
and  m^  ;  fa  into  nz  and  m2  ;  and  /3  into  ns  and  m3.  The  forces  in 
one  axis  are  w1?  n^  and  ns  ;  and  those  in  the  other  m^  mz,  and  ms. 
And  by  the  parallelogram  of  forces  it  is  well  known  that 


if  we  take  in  the  axis  OX  any  point  P,  and  let  fall  from  it 


MECHANICAL   POWERS.  425 

the  perpendiculars  PA,  PB,  PC,  on  the  directions  of  the  forces  ft, 
/3,/3,  we  obtain  the  following  similar  right-angled  triangles,  namely, 

GAP  and  0  n±f±  are  similar  ; 
OBP  and  0  nafs  --  ; 
OOP  and(K/3  -  ; 
Ow        OA 


t  ^ 

=  j  and  nt  =  "QP/I*     I*  1S  easily  seen  also  that 

CO  _  *    BO  ^ 

Wfl  =  "OF*'3  ;  and  ^3  =  OF*'  3' 

If  the  values  be  substituted  in  (E),  we  obtain 
BO  x/3=CO  x/2+  AO  x/a. 

From  the  similarity  of  these  triangles,  and  the  remaining  equa- 
tion of  (E),  we  can  readily  find  that 

PB  x  /8  =  PA  x  ft  +  PC  x  /£. 
The  equation  becomes  more  compact  by  putting 

OA,  OC,  OB,  respectively  equal  «4,  sa  s3  ;  and 
PA,  PC,  PB,    ----  qi,  qz,  ga. 


The  same  holds  good  with  any  number  of  forces  /±,  /3,  /3,  &c., 
and  their  resultant  /M,  that  is 


J  +  /3?3  +  &c. 

If  the  point  of  application  0,  move  in  a  straight  line  to  P,  then 
OA  =  st  is  called  the  space  of  the  force  /t,  and/^  the  work  done 
by  the  force  /1?  in  moving  the  body  from  0  to  P.  OB  is  the  space 
of  the  resultant,  and  the  product  /3*3,  the  work  done  by  it.  fas2 
is  the  work  done  by/3  in  moving  the  material  point  0  from  0  to  P. 
Hence  the  work  done  by  the  resultant  is  equal  to  all  the  work  done 
by  the  component  forces,  as  we  have  shown, 


PRINCIPLES  AND  PRACTICAL  APPLICATIONS  OF 
MECHANICAL  POWERS. 

MECHANICAL  Powers,  or  the  Elements  of  Machinery,  are  certain 
simple  mechanical  arrangements  whereby  weights  may  be  raised  or 
resistances  overcome  with  the  exertion  of  less  power  or  strength 
than  is  necessary  without  them. 

They  are  usually  accounted  six  in  number,  viz.  the  lever,  the 
wheel  and  axle,  the  pulley,  the  inclined  plane,  the  wedge,  and  the 
screw;  but  properly  two  of  these  comprise  the  whole,  namely,  the 
lever  and  inclined  plane, — the  wheel  and  axle  being  only  a  lever 
of  the  first  kind,  and  the  pulley  a  lever  of  the  second, — the  wedge 
and  the  screw  being  also  similarly  allied  to  that  of  the  inclined 
plane :  however,  although  such  seems  to  be  the  case  in  these  re- 


426  THE  PRACTICAL  MODEL  CALCULATOR. 

spects,  yet  they  each  require,  on  account  of  their  various  modifica- 
tions, a  peculiar  rule  of  estimation  adapted  expressly  to  the  differ- 
ent circumstances  in  which  they  are  individually  required  to  act. 

THE  LEVER. 

Levers,  according  to  mode  of  application,  as  the  following,  are 
distinguished  as  be- 
ing of  the  first,  se-  2nd  T aq-        3rd. 

cond,  or  third  kind;     71ha  [  £  4481bs. 

and  although  levers 
of  equal  lengths  pro- 
duce different  ef- 
fects, the  general 
principles  of  esti- 
mation in  all  are 
the  same ;  namely, 
the  power  is  to  the  56  lbs-  66  lbs- 

weight  or  resistance,  as  the  distance  of  the  one  end  to  the  fulcrum 
is  to  the  distance  of  the  other  end  to  the  same  point. 

In  the  first  kind,  the  power  is  to  the  resistance,  as  the  distance 
AB  is  to  the  distance  BC. 

In  the  second,  the  power  is  to  the  resistance,  as  the  distance  AB 
is  to  that  of  AC  ;  and, 

In  the  third,  the  resistance  is  to  the  power,  as  the  distance  AB 
is  to  that  of  AC. 

RULE,  first  kind. — Divide  the  longer  by  the  shorter  end  of  the 
lever  from  the  fulcrum,  and  the  quotient  is  the  effective  force  that 
the  power  applied  is  equal  to. 

Let  the  handle  of  a  pump  equal  65  inches  in  length,  and  10 
inches  from  the  shortest  end  to  centre  of  motion ;  what  is  the 
amount  of  effective  leverage  thereby  obtained  ? 

65  -  10  =  55,  and  ^  =  5J  to  1. 

Required  the  situation  of  the  fulcrum  on  which  to  rest  a  lever 
of  15  feet,  so  that  2J  cwt.  placed  at  one  end  may  equipoise  30  cwt. 
at  the  other,  the  weight  of  the  lever  not  being  taken  into  account. 

15  x  2*5 

TJ-= — r~vf\  —  1*154  feet  from  the  end  on  which  the  30  cwt.  is  to 

A'O    +    OU 

be  placed. 

It  is  by  the  second  kind  of  lever  that  the  greatest  effect  is  ob- 
tained from  any  given  amount  of  power ;  hence  the  propriety  of 
the  application  of  this  principle  to  the  working  of  force  pumps,  and 
shearing  of  iron,  as  by  the  lever  of  a  punching-press,  &c. 

RULE,  second  kind. — Divide  the  whole  length  of  lever,  or  dis- 
tance from  power  to  fulcrum,  by  the  distance  from  fulcrum  to 
weight,  and  the  quotient  is  the  proportion  of  effect  that  the  power 
is  to  the  weight  or  resistance  to  be  overcome. 

Required  the  amount  of  effect  or  force  produced  by  a  power  of 


MECHANICAL    POWERS.  427 

50  Ibs.  on  the  ram  of  a  Bramah's  pump,  the  length  of  the  lever 
being  3  feet,  and  distance  from  ram  to  fulcrum  4J  inches. 

Q£» 

3  feet  =  36  inches,  and  ^  =  8,  or  the  power  and  resistance 

are  to  each  other  as  8  to  1 ;  hence  50  X  8  =  400  Ibs.  force  upon 
the  ram. 

The  lever  on  the  safety  valve  of  a  steam  boiler  is  of  the  third 
kind,  the  action  of  the  steam  being  the  power,  and  the  weight  or 
spring-balance  attached  the  resistance  ;  but  in  such  application  the 
action  of  the  lever's  weight  must  also  be  taken  into  account. 

THE   WHEEL  AND   PINION,   OR   CRANE. 

The  mechanical  advantage  of  the  wheel  and  axle,  or  crane,  is  as 
the  velocity  of  the  weight  to  the  velocity  of  the  power ;  and  being 
only  a  modification  of  the  first  kind  of  lever,  it  of  course  partakes 
of  the  same  principles. 

RULE. —  To  determine  the  amount  of  effective  power  produced 
from  a  given  power  by  means  of  a  crane  with  known  peculiarities. — 
Multiply  together  the  diameter  of  the  circle  described  by  the  winch, 
or  handle,  and  the  number  of  revolutions  of  the  pinion  to  1  of  the 
wheel ;  divide  the  product  by  the  barrel's  diameter  in  equal  terms 
of  dimensions,  and  the  quotient  is  the  effective  power  to  1  of  ex- 
ertive  force. 

Let  there  be  a  crane  the  winch  of  which  describes  a  circle  of  30 
inches  in  diameter ;  the  pinion  makes  8  revolutions  for  1  of  the 
wheel,  and  the  barrel  is  11  inches  in  diameter;  required  the  effec- 
tive power  in  principle,  also  the  weight  that  36  Ibs.  would  raise, 
friction  not  being  taken  into  account. 

30  x  8 

n      =  21-8  to  1  of  exertive  force ;  and  21-8  X  36  =  784-8  Ibs. 

RULE. — G-iven  any  two  parts  of  a  crane,, to  find  the  third,  that 
shall  produce  any  required  proportion  of  mechanical  effect. — Mul- 
tiply the  two  given  parts  together,  and  divide  the  product  by  the 
required  proportion  of  effect ;  the  quotient  is  the  dimensions  of  the 
other  parts  in  equal  terms  of  unity. 

Suppose  that  a  crane  is  required,  the  ratio  of  power  to  effect 
being  as  40  to  1,  and  that  a  wheel  and  pinion  11  to  1  is  unavoid- 
ably compelled  to  be  employed,  also  the  throw  of  each  handle  to 
be  16  inches ;  what  must  be  the  barrel's  diameter  on  which  the 
rope  or  chain  must  coil  ? 

16  x  2  =  32  inches  diameter  described  by  the  handle. 
And  — JQ —  =  8*8  inches,  the  barrel's  diameter. 

THE   PULLEY. 

The  principle  of  the  pulley,  or,  more  practically,  the  block  and 
tackle,  is  the  distribution  of  weight  on  various  points  of  support ; 
the  mechanical  advantage  derived  depending  entirely  upon  the 


428  THE   PRACTICAL   MODEL   CALCULATOR. 

flexibility  and  tension  of  the  rope,  and  the  number  of  pulleys  or 
sheives  in  the  lower  or  rising  block  :  hence,  by  blocks  and  tackle  of 
the  usual  kind,  the  power  is  to  the  weight  as  the  number  of  cords 
attached  to  the  lower  block  ;  whence  the  following  rules. 

Divide  the  weight  to  be  raised  by  the  number  of  cords  leading 
to,  from,  or  attached  to  the  lower  block  ;  and  the  quotient  is  the 
power  required  to  produce  an  equilibrium,  provided  friction  did  not 
exist. 

Divide  the  weight  to  be  raised  by  the  power  to  be  applied  ;  the 
quotient  is  the  number  of  sheives  in,  or  cords  attached  to  the  rising 
block. 

Required  the  power  necessary  to  raise  a  weight  of  3000  Ibs.  by 
a  four  and  five-sheived  block  and  tackle,  the  four  being  the  mov- 
able or  rising  block. 

Necessarily  there  are  nine  cords  leading  to  and  from  the  rising 
block. 

Consequently  —  g—  =  333  Ibs.,  the  power  required. 

I  require  to  raise  a  weight  of  1  ton  18  cwt.,  or  4256  Ibs.  ;  the 
amount  of  my  power  to  effect  this  object  being  500  Ibs.,  what  kind 
of  block  and  tackle  must  I  of  necessity  employ  ? 

4256 

8'51  cords  ;  of  necessity  there  must  be  4  sheives  or  9 


cords  in  the  rising  block. 

As  the  effective  power  of  the  crane  may,  by  additional  wheels 
and  pinions,  be  increased  to  any  required  extent,  so  may  the  pulley 
and  tackle  be  similarly  augmented  by  purchase  upon  purchase. 

THE   INCLINED   PLANE. 

The  inclined  plane  is  properly  the  second  elementary  power,  and 
may  be  defined  the  lifting  of  a  load  by  regular  instalments.  In 
principle  it  consists  of  any  right  line  not  coinciding  with,  but  ly- 
ing in  a  sloping  direction  to,  that  of  the  horizon  ;  the  standard  of 
comparison  of  which  commonly  consists  in  referring  the  rise  to  so 
many  parts  in  a  certain  length  or  distance,  as  1  in  100,  1  in  200, 
&c.,  —  the  first  number  representing  the  perpendicular  height,  and 
the  latter  the  horizontal  length  in  attaining  such  height,  both  num- 
bers being  of  the  same  denomination,  unless  otherwise  expressed  ; 
but  it  may  be  necessary  to  remark,  that  the  inclination  of  a  plane, 
the  sine  of  inclination,  the  height  per  mile,  or  the  height  for  any 
length,  the  ratio,  &c.,  are  all  synonymous  terms. 

The  advantage  gained  by  the  inclined  plane,  when  the  power  acts 
in  a  parallel  direction  to  the  plane,  is  as  the  length  to  the  height 
or  angle  of  inclination  :  hence  the  rule.  Divide  the  weight  by  the 
ratio  of  inclination,  and  the  quotient  equal  the  power  that  will  just 
support  that  weight  upon  the  plane.  Or,  multiply  the  weight  by 
the  height  of  the  plane,  and  divide  by  the  length,  —  the  quotient  is 
the  power. 


MECHANICAL    POWERS. 


429 


Required  the  power  or  equivalent  weight  capable  of  supporting 
a  load  of  350  Ibs.  upon  a  plane  of  1  in  12,  or  3  feet  in  height  and 
36  feet  in  length. 

-jg-  =  29-16  Ibs.,  or  -—§§ —  =  29-16  Ibs.  power,  as  before. 

The  weight  multiplied  by  the  length  of  the  base,  and  the  product 
divided  by  the  length  of  the  incline,  the  quotient  equal  the  pres- 
sure or  downward  weight  upon  the  incline. 

TABLE  showing  the  Resistance  opposed  to  the  Motion  of  Carriages 
on  different  Inclinations  of  Ascending  or  Descending  Planes, 
whatever  part  of  the  insistent  weight  they  are  drawn  by. 


4 

1 

HUNDREDS. 

100 

200 

300 

400 

500 

600 

700 

800 

900 

•01 

•005 

•00333 

•0025 

•002 

•00167 

•00143 

•00125 

•00111 

10 

•1 

•00909 

•00476 

•00322 

•00244 

•00196 

•00164 

•00141 

•00123 

•0011 

20 

•05 

•00833 

•00454 

•00312 

•00238 

•00192 

•00161 

•00139 

•00122 

•00109 

30 

•0333 

•00769 

•00435 

•00303 

•00232 

•00189 

•00159 

•00137 

•0012 

•00107 

40 

•025 

•00714 

•00417 

•00294 

•00227 

•00185 

•00156 

•00135 

•00119 

•00106 

50 

•02 

•00667 

•004 

•00286 

•00222 

•00182 

•00154 

•00133 

•00118 

•00105 

60 

•0166 

•00625 

•00385 

•00278 

•00217 

•00178 

•00151 

•00131 

•00116 

•00104 

70 

•0143 

•00588 

•0037 

•0027 

•00213 

•00175 

•00149 

•0013 

•00115 

•00103 

80 

•0125 

•00555 

•00357 

•00263 

•00208 

•00172 

•00147 

•00128 

•00114 

•00102 

90 

•0111 

•00526 

•00345 

•00256 

•00204 

•00169 

•00145 

•00126 

•00112 

•00101 

Although  this  table  has  been  calculated  particularly  for  car- 
riages on  railway  inclines,  it  may  with  equal  propriety  be  applied 
to  any  other  incline,  the  amount  of  traction  on  a  level  being  known. 

Application  of  the  preceding  Table. 

What  weight  will  a  tractive  power  of  150  Ibs.  draw  up  an  incline 
of  1  in  340,  the  resistance  on  the  level  being  estimated  at  ^th 
part  of  the  insistent  weight  ? 

In  a  line  with  40  in  the  left-hand  column  and  under  200  is  '00417 
Also  in  the  same  line  and  under  390  is  ........................   -00294 

Added  together  =  -00711 
^  21097  Ibs.  weight  drawn  up  the  plane. 


150 


Then 

What  weight  would  a  force  of  150  Ibs.  draw  down  the  same  plane, 
the  fraction  on  the  level  being  the  same  as  before  ? 

Friction  on  the  level  =  -00417 
Gravity  of  the  plane  =  -00294  subtract 

=  -00123 
And  .QQ-ioo  —  121915  Ibs.  weight  drawn  down  the  plane. 

Example  of  incline  when  velocity  is  taken  into  account.  —  A  power 
of  230  Ibs.,  at  a  velocity  of  75  feet  per  minute,  is  to  be  employed 
for  moving  weights  up  an  inclined  plane  12  feet  in  height  and  163 


430 


THE   PRACTICAL   MODEL   CALCULATOR. 


feet  in  length,  the  least  velocity  of  the  weight  to  be  8  feet  per 
minute ;  required  the  greatest  weight  that  the  power  is  equal  to. 

230  x  75  x  163       2811750 

=  29288  Ibs.,  or  13-25  tons. 


12  x  8 


96 


TABLE  of  Inclined  Planes,  showing  the  ascent  or  descent  per  yard, 
and  the  corresponding  ascent  or  descent  per  chain,  per  mile;  and 
also  the  ratio. 


Per  yard. 

Per  chain. 

Per  mile. 

Ratio. 

Per  yard. 

Per  chain. 

Per  mile. 

Ratio. 

In  parts 
of  an  in. 

In  dec'ls. 
of  an  inch. 

Inches. 

Feet. 

1  inch. 

In  parts 
of  an  in. 

In  decimals 
of  an  inch. 

Inches. 

Feet. 

linchj 

& 

•0156 

•344 

2-29 

2304 

T7S 

•4375 

9-625 

64-17 

82 

*"l 

•0208 

•458 

3-06 

1728 

? 

•5 

11 

73-33 

72 

sV 

•0312 

•687 

4-58 

1152 

A 

•5625 

12-375 

82-5 

64 

& 

•0417 

•917 

6-11 

864 

ff 

•5833 

12-833 

85-56 

62 

iV 

•0625 

1-375 

9-17 

576 

1 

•6 

13-2 

88 

60 

TV 

•0833 

1-833 

12-22 

432 

•625 

13-75 

91-67 

58 

A 

•1 

2*2 

14-67 

360 

f 

•6667 

14-667 

97-78 

54 

i 

•125 

2-75 

18-33 

288 

H 

•6875 

15-125 

100-83 

52 

* 

•1667 

3-667 

24-44 

216 

iV 

•7 

15-4 

102-67 

51 

ft 

•1875 

4-125 

27-50 

192 

i 

•7.5 

16-5 

110 

48 

i 

*2 

4-4 

29-33 

180 

I 

•8 

17-6 

117-33 

45 

•25 

5-5 

36-67 

144 

if 

•8125 

17-875 

119-17 

44 

A 

•3 

6-6 

44 

120 

£ 

•8333 

18-333 

122-22 

43 

•3125 

6-875 

45-83 

115 

1 

•875 

19-25 

128-33 

41 

•3333 

7-333 

48-89 

108 

T90 

•9 

19-8 

132 

40 

•375 

8-25 

55 

96 

tt 

•9167 

20-167 

134-44 

39 

I 

•4 

8-8 

58-67 

20 

if 

•9375 

20-625 

137-5 

38 

A 

•4167 

9-167 

61-11 

86 

1 

1 

22 

146-67 

36 

THE   WEDGE. 

The  wedge  is  a  double  inclined  plane  ;  consequently  its  principles 
are  the  same  :  hence,  when  two  bodies  are'  forced  asunder  by  means 
of  the  wedge  in  a  direction  parallel  to  its  head, — Multiply  the  re- 
sisting power  by  half  the  thickness  of  the  head  or  back  of  the  wedge, 
and  divide  the  product  by  the  length  of  one  of  its  inclined  sides ; 
the  quotient  is  the  force  equal  to  the  resistance. 

The  breadth  of  the  back  or  head  of  a  wedge  being  3  inches,  and 
its  inclined  sides  each  10  inches,  required  the  power  necessary  to 
act  upon  the  wedge  so  as  to  separate  two  substances  whose  resist- 
ing force  is  equal  to  150  Ibs. 

150  x  1-5 
-JQ—  -  =  22-5  Ibs. 

When  only  one  of  the  bodies  is  movable,  the  whole  breadth  of 
the  wedge  is  taken  for  the  multiplier. 

THE  SCREW. 

The  screw,  in  principle,  is  that  of  an  inclined  plane  wound  around 
a  cylinder,  which  generates  a  spiral  of  uniform  inclination,  each 
revolution  producing  a  rise  or  traverse  motion  equal  to  the  pitch 
of  the  screw,  or  distance  between  two  consecutive  threads, — the 
pitch  being  the  height  or  angle  of  inclination,  and  the  circumference 


MECHANICAL    POWERS.  431 

the  length  of  the  plane  when  a  lever  is  not  applied  ;  but  the  lever 
being  a  necessary  qualification  of  the  screw,  the  circle  which  it  de- 
scribes is  taken,  instead  of  the  screw's  circumference,  as  the  length 
of  the  plane  :  hence  the  mechanical  advantage  is,  as  the  circum- 
ference of  the  circle  described  by  the  lever  where  the  power  acts, 
is  to  the  pitch  of  the  screw,  so  is  the  force  to  the  resistance  in 
principle. 

Required  the  effective  power  obtained  by  a  screw  of  |  inch  pitch, 
and  moved  by  a  force  equal  to  50  Ibs.  at  the  extremity  of  a  lever 
30  inches  in  length. 

30  X  «  x  8-1416  x  50 


Required  the  power  necessary  to  overcome  a  resistance  equal  to 
7000  Ibs.  by  a  screw  of  1J  inch  pitch,  and  moved  by  a  lever  25 
inches  in  length. 

7000  x  1-25 

25  x  2  x  8-1416  =  55'78  lbs'  P°wer' 

In  the  case  of  a  screw  acting  on  the  periphery  of  a  toothed  wheel, 
the  power  is  to  the  resistance,  as  the  product  of  the  circle's  circum- 
ference described  by  the  winch  or  lever,  and  radius  of  the  wheel, 
to  the  product  of  the  screw's  pitch,  and  radius  of  the  axle,  or  point 
whence  the  power  is  transmitted  ;  but  observe,  that  if  the  screw 
consist  of  more  than  one  helix  or  thread,  the  apparent  pitch  must 
be  increased  so  many  times,  as  there  are  threads  in  the  screw. 
Hence,  to  find  what  weight  a  given  power  will  equipoise  : 

RULE.  —  Multiply  together  the  radius  of  the  wheel,  the  length  of 
the  lever  at  which  the  power  acts,  the  magnitude  of  the  power,  and 
the  constant  number  6-2832  ;  divide  the  product  by  the  radius  of 
the  axle  into  the  pitch  of  the  screw,  and  the  quotient  is  the  weight 
that  the  power  is  equal  to. 

What  weight  will  be  sustained  in  equilibrio  by  a  power  of  100 
lbs.  acting  at  the  end  of  a  lever  24  inches  in  length,  the  radius  of 
the  axle,  or  point  whence  the  power  is  transmitted,  being  8  inches, 
the  radius  of  the  wheel  14  inches,  the  screw  consisting  of  a  double 
thread,  and  the  apparent  pitch  equal  f  of  an  inch  ? 
14  x  24  x  100  x  6-2832 

•625  x  2  x  8""     ~  =  21111'55  lbs->  or  9'4  tons>  the 
.  power  sustained. 

If  an  endless  screw  be  turned  by  a  handle  of  20  inches,  the  threads 

of  the  screw  being  distant  half  an  inch  ;  the  screw  turns  a  toothed 

wheel,  the  pinion  of  which  turns  another  wheel,  and  the  pinion  of 

this  another  wheel,  to  the  barrel  of  which  a  weight  W  is  attached  ; 

it  is  required  to  find  the  weight  a  man  will  be  able  to  sustain,  who 

acts  at  the  handle  with  a  force  of  150  lbs.,  the  diameters  of  the 

wheels  being  18  inches,  and  those  of  the  pinions  and  barrel  2  inches. 

150  x  20  x  3-1416  x  2  x  183  =  W  x  23  x  £  ; 

.-.  W  =  12269  tons. 


432          THE  PRACTICAL  MODEL  CALCULATOR. 


CONTINUOUS  CIRCULAR  MOTION. 

IN  mechanics,  circular  motion  is  transmitted  by  means  of  ivheels, 
drums,  or  pulleys;  and  accordingly  as  the  driving  and  driven  are 
of  equal  or  unequal  diameters,  so  are  equal  or  unequal  velocities 
produced  :  hence  the  principle  on  which  the  following  rules  are 
founded. 

RULE.  —  When  time  is  not  taken  into  account.  —  Divide  the  greater 
diameter,  or  number  of  teeth,  by  the  lesser  diameter,  or  number 
of  teeth,  and  the  quotient  is  the  number  of  revolutions  the  lesser 
will  make  for  1  of  the  greater. 

How  many  revolutions  will  a  pinion  of  20  teeth  make  for  1  of  a 
wheel  with  125  ? 

125  -*-  20  =  6-25,  or  6J  revolutions. 

Intermediate  wheels,  of  whatever  diameters,  so  as  to  connect 
communication  at  any  required  distance  apart,  cause  no  variation 
of  velocity  more  than  otherwise  would  result  were  the  first  and  last 
in  immediate  contact. 

RULE.  —  To  find  the  number  of  revolutions  of  the  last,  to  1  of  the 
first,  in  a  train  of  wheels  and  pinions.  —  Divide  the  product  of  all 
the  teeth  in  the  driving,  by  the  product  of  all  the  teeth  in  the 
driven,  and  the  quotient  equal  the  ratio  of  velocity  required. 

Required  the  ratio  of  velocity  of  the  last,  to  1  of  the  first,  in  the 
following  train  of  wheels  and  pinions  ;  viz.,  pinions  driving,  —  the 
first  of  which  contains  10  teeth,  the  second  15,  and  third  18  ;  — 
wheels  driven,  —  first  15  teeth,  second  25,  and  third  32. 

10x1^x18 

^  a  rev°luti°n  tne  wheel  will  make  to  1 


15  X  25  X  32 

of  the  pinion. 

A  wheel  of  42  teeth  giving  motion  to  one  of  12,  on  which  shaft 
is  a  pulley  of  21  inches  diameter,  driving  one  of  6  ;  required  tJie 
number  of  revolutions  of  the  last  pulley  to  1  of  the  first  wheel. 

42  x  21 
^2  x  Q  —  12*25,  or  12J  revolutions. 

Where  increase  or  decrease  of  velocity  is  required  to  be  commu- 
nicated by  wheel-work,  it  has  been  demonstrated  that  the  number 
of  teeth  on  each  pinion  should  not  be  less  than  1  to  6  of  its"  wheel, 
unless  there  be  some  other  important  reason  for  a  higher  ratio. 

RULE.  —  When  time  must  be  regarded.  —  Multiply  the  diameter, 
or  number  of  teeth  in  the  driver,  by  its  velocity  in  any  given  time, 
and  divide  the  product  by  the  required  velocity  of  the  driven  ;  the 
quotient  equal  the  number  of  teeth,  or  diameter  of  the  driven,  to 
produce  the  velocity  required. 

If  a  wheel  containing  84  teeth  makes  20  revolutions  per  minute, 
how  many  must  another  contain  to  work  in  contact,  and  make  60 
revolutions  in  the  same  time  ? 


CONTINUOUS    CIRCULAR   MOTION.  433 

84  x  20 


28  teeth. 


60 

From  a  shaft  making  45  revolutions  per  minute,  and  with  a  pinion 
9  inches  diameter  at  the  pitch  line,  I  wish  to  transmit  motion  at  15 
revolutions  per  minute  ;  what  at  the  pitch  line  must  be  the  diameter 
of  the  wheel  ? 

45  x  9 

— jg —  =  27  inches. 

Required  the  diameter  of  a  pulley  to  make  16  revolutions  in  the 
same  time  as  one  of  24  inches  making  36. 

24x36 

— jg —  =  54  inches. 

RULE. —  The  distance  between  the  centres  and  velocities  of  two 
wheels  being  given,  to  find  their  proper  diameters. — Divide  the 
greatest  velocity  by  the  least ;  the  quotient  is  the  ratio  of  diameter 
the  wheels  must  bear  to  each  other.  Hence,  divide  the  distance 
between  the  centres  by  the  ratio  plus  1 ;  the  quotient  equal  the 
radius  of  the  smaller  wheel ;  and  subtract  the  radius  thus  obtained 
from  the  distance  between  the  centres ;  the  remainder  equal  the 
radius  of  the  other. 

The  distance  of  two  shafts  from  centre  to  centre  is  50  inches, 
and  the  velocity  of  the  one  25  revolutions  per  minute,  the  other  is 
to  make  80  in  the  same  time ;  the  proper  diameters  of  the  wheels 
at  the  pitch  lines  are  required. 

80  -f-  25  =  3*2,  ratio  of  velocity,  and  Q.Q  4.  i  =  11*9,  the  ra- 
dius of  the  smaller  wheel ;  then  50  —  11-9  =  38-1,  radius  of  larger ; 
their  diameters  are  11-9  X  2  =  23-8,  and  38-1  X  2  =  76-2  inches. 

To  obtain  or  diminish  an  accumulated  velocity  by  means  of 
wheels  and  pinions,  or  wheels,  pinions,  and  pulleys,  it  is  necessary 
that  a  proportional  ratio  of  velocity  should  exist,  and  which  is 
simply  thus  attained  : — Multiply  the  given  and  required  velocities 
together,  and  the  square  root  of  the  product  is  the  mean  or  propor- 
tionate velocity. 

Let  the  given  velocity  of  a  wheel  containing  54  teeth  equal  16 
revolutions  per  minute,  and  the  given  diameter  of  an  intermediate 
pulley  equal  25  inches,  to  obtain  a  velocity  of  81  revolutions  in  a 
machine ;  required  the  number  of  teeth  in  the  intermediate  wheel, 
and  diameter  of  the  last  pulley. 

v/81  x  16  =  36  mean  velocity. 

54  x  16  25  x  36 

— gg —  =  24  teeth,  and  — gy —  =  11  •!  inches,  diameter  of 

pulley. 

To  determine  the  proportion  of  wheels  for  screw  cutting  by  a 
lathe. — In  a  lathe  properly  adapted,  screws  to  any  degree  of  pitch, 
or  number  of  threads  in  a  given  length,  may  be  cut  by  means  of  a 

28  \ 


434          THE  PRACTICAL  MODEL  CALCULATOR. 

leading  screw  of  any  given  pitch,  accompanied  with  change  wheels 
and  pinions  ;  course  pitches  being  effected  generally  by  means  of 
one  wheel  and  one  pinion  with  a  carrier^  or  intermediate  wheel, 
which  cause  no  variation  or  change  of  motion  to  take  place  :  hence 
the  following 

RULE.  —  Divide  the  number  of  threads  in  a  given  length  of  the 
screw  which  is  to  be  cut,  by  the  number  of  threads  in  the  same 
length  of  the  leading  screw  attached  to  the  lathe  ;  and  the  quotient 
is  the  ratio  that  the  wheel  on  the  end  of  the  screw  must  bear  to 
that  on  the  end  of  the  lathe  spindle. 

Let  it  be  required  to  cut  a  screw  with  5  threads  in  an  inch,  the 
leading  screw  being  of  J  inch  pitch,  or  containing  2  threads  in  an 
inch  ;  what  must  be  the  ratio  'of  wheels  applied  ? 

5  -j-  2  =  2-5,  the  ratio  they  must  bear  to  each  other. 
Then  suppose  a  pinion  of  40  teeth  be  fixed  upon  for  the  spindle,  — 
40  X  2*5  =  100  teeth  for  the  wheel  on  the  end  of  the  screw. 

But  screws  of  a  greater  degree  of  fineness  than  about  8  threads 
in  an  inch  are  more  conveniently  cut  by  an  additional  wheel  and 
pinion,  because  of  the  proper  degree  of  velocity  being  more  effec- 
tively attained  ;  and  these,  on  account  of  revolving  upon  a  stud, 
are  commonly  designated  the  stud-wheels,  or  stud-wheel  and  pinion  ; 
but  the  mode  of  calculation  and  ratio  of  screw  are  the  same  as  in 
the  preceding  rule  ;  —  hence,  all  that  is  further  necessary  is  to  fix 
upon  any  3  wheels  at  pleasure,  as  those  for  the  spindle  and  stud- 
wheels,  —  then  multiply  the  number  of  teeth  in  the  spindle-wheel 
by  the  ratio  of  the  screw,  and  by  the  number  of  teeth  in  that  wheel 
or  pinion  which  is  in  contact  with  the  wheel  on  the  end  of  the  screw  ; 
divide  the  product  by  the  stud-wheel  in  contact  with  the  spindle- 
wheel,  and  the  quotient  is  the  number  of  teeth  required  in  the  wheel 
on  the  end  of  the  leading  screw. 

Suppose  a  screw  is  required  to  be  cut  containing  25  threads  in 
an  inch,  the  leading  screw  as  before  having  2  threads  in  an  inch, 
and  that  a  wheel  of  60  teeth  is  fixed  upon  for  the  end  of  the  spin- 
dle, 20  for  the  pinion  in  contact  with  the  screw-wheel,  and  100  for 
that  in  contact  with  the  wheel  on  the  end  of  the  spindle  ;  —  required 
the  number  of  teeth  in  the  wheel  for  the  end  of  the  leading 
screw. 

25  -4-  2  =  12-5,  and  -  >  —  =  150  teeth. 


Or,  suppose  the  spindle  and  screw-wheels  to  be  those  fixed  upon, 
also  any  one  of  the  stud-  wheels,  to  find  the  number  of  teeth  in  the 
other. 

60  x  12-5  60  x  12-5  x  20 

150  x  100  =  2Q  teeth>  or  "      150-     —100  teeth. 


CONTINUOUS   CIRCULAR   MOTION. 


435 


TABLE  of  Change  Wheels  for  Screiv  Cutting,  the  leading  screw 
being  of  J  inch  pitch,  or  containing  two  threads  in  an  inch. 


jr  of  threads  in 
)f  screw. 

Number  of 
teeth  in 

a 

Number  of  teeth  in 

er  of  threads  in 
rf  screw. 

Number  of  teeth  in 

, 

J 
jj 

"".S 

k 
\ 

i 

•3 

.2 

in  contact 
spindle-wheel. 

.  in  contact 
screw-wheel. 

ig  screw- 

1 

2 

•sl 

v^ 

„  0 

I 

!| 

li 

|| 

II 

il 

|| 

I~ 

.2  73 

fj 

S-S 

II 

I| 

25 

P 

jg.S 

3^ 

$'f 

jg.S 

3* 

£* 

£* 

J^ 

&•" 

3* 

^? 

S^ 

1 

80 

40 

8i 

40 

55 

20 

60 

19 

50 

95 

20 

100 

li 

80 

50 

gl 

90 

85 

20 

90 

19A 

80 

120 

20 

130 

1£ 

80 

60 

8| 

60- 

70 

20 

75 

20 

60 

100 

20 

120 

If 

80 

70 

9} 

90 

90 

20 

95 

20^ 

40 

90 

20 

90 

2 

80 

90 

9| 

40 

60 

20 

65 

21 

80 

120 

20 

140 

21 

80 

90 

10 

60 

75 

20 

80 

22 

60 

110 

20 

120 

2! 

80 

100 

10i 

50 

70 

20 

75 

22£ 

80 

120 

20 

150 

2| 

80 

110 

11 

60 

55 

20 

120 

22| 

80 

130 

20 

140 

3 

80 

120 

12 

90 

90 

20 

120 

23f 

40 

95 

20 

100 

3J 

80 

130 

12f 

60 

85 

20 

90 

24 

65 

120 

20 

130 

3£ 

80 

140 

13 

90 

90 

20 

130 

25 

60 

100 

20 

150 

33. 

80 

150 

13J 

60 

90 

20 

90 

25i 

30 

85 

20 

90 

44 

40 

80 

13| 

80 

100 

20 

110 

26 

70 

130 

20 

140 

4-1- 

40 

85 

14 

90 

90 

20 

140 

27 

40 

90 

20 

120 

4J 

40 

90 

14J 

60 

90 

20 

95 

27* 

40 

100 

20 

110 

4J 

40 

95 

15 

90 

90 

20 

150 

28 

75 

140 

20 

150 

5 

40 

100 

16 

60 

80 

20 

120 

28J 

30 

90 

20 

95 

5j 

40 

110 

16J 

80 

100 

20 

130 

30 

70 

140 

20 

150 

6 

40 

120 

16J 

80 

110 

20 

120 

32 

30 

80 

20 

120 

6^- 

40 

130 

17 

45 

85 

20 

'  90 

33 

40 

110 

20 

120 

7 

40 

140 

171 

80 

100 

20 

140 

34 

30 

85 

20 

120 

7£ 

40 

150 

18 

40 

60 

20 

120 

35 

60 

140 

20 

150 

8 

30 

120 

18f 

80 

100 

20 

150 

36 

30 

90 

20 

120 

TABLE  ty  which  to  determine  the  Number  of  Teeth,  or  Pitch  of 

Small  Wheels. 


Diametral 
pitch. 

Circular 
pitch. 

Diametral 
pitch. 

Circular 

pitch. 

3 

1-047 

9 

•349 

4 

•785 

10 

•314 

5 

•628 

12 

•262 

6 

•524 

14 

•224 

7 

•449 

16 

•196 

8 

•393 

20 

•157 

Required  the  number  of  teeth  that  a  wheel  of  16  inches  diameter 
will  contain  of  a  10  pitch. 

16  X  10  =  160  teeth,  and  the  circular  pitch  =  *314  inch. 

What  must  be  the  diameter  of  a  wheel  for  a  9  pitch  of  126  teeth  ? 

126 

14  inches  diameter,  circular  pitch  *349  inch. 


The  pitch  is  reckoned  on  the  diameter  of  the  wheel  instead  of 
the  circumference,  and  designated  wheels  of  8  pitch,  12  pitch,  &c. 


436 


THE  PRACTICAL  MODEL  CALCULATOR. 


TABLE  of  the  Diameters  of  Wheels  at  their  pitch  circle,  to  contain 
a  required  number  of  teeth  at  a  given  pitch. 


« 

PITCH  OF  THE  TEETH  IN  INCHES. 

in. 

If 

li    |   If 

14  |  li  |  11 

13 

2  in. 

24 

2i 

24 

22   |  3  in. 

DIAMETER  AT  THE  FITCH  CIRCLE   IN  FEET  AND  INCHES. 

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CONTINUOUS   CIRCULAR   MOTION. 


437 


Number  of 
teeth. 

PITCH  OF  THE  TEETH  IN  INCHES. 

lie. 

H 

li    |   If 

14 

If 

IS 

13 

2  in. 

24 

2i 

24 

2f    |3in. 

DIAMETER   AT  THE   PITCH  CIRCLE   IN   FEET  AND   INCHES. 

71 

llOf 

2    142    4i 

2    7 

2    933    OS 

3    34 

3    6g 

3    9i 

4    0 

4    23|4    84|5    24 

5    7f 

72 

1103 

2    1S'2    4f 

2    74 

210| 

3    li 

3    44 

3    6S 

3    93 

4    OS 

4    3414    9i  5    3 

5    8f 

73 

nil 

2    242    5 

2    8 

2  103 

3    IS 

3    4f 

3    74 

3  104 

4    If 

4    4i!4  10 

5    33 

5    9f 

74 

1114 

2    2412    54 

2    8f 

2  llf 

3    2i 

3    5i 

3    73 

3114 

4    2 

4    5    4  103  5    4f 

5  lOf 

75 

1113 

2    2312    53 

2    83J2  113<3    2S 

3    5S 

3    8S 

3  11| 

4    2| 

4    5| 

4  llf  i5    5f 

5  llf 

762    04 

2    3i2    6i 

2    9i  3    Oi  3    3i 

3    6f  3    9f 

4    Of 

4    34 

4    64 

5    0415    64 

6    04 

772    04 

2    34  2    6| 

2    9f  3    0| 

3    33 

3    63|3    93 

4    1 

4    4 

4    74 

5    li!5    7f 

6    14 

782    03 

2    3i  2    7 

2  104  3    14J3    4g 

3    74 

3  104 

'4    If 

4    45 

4    73|5    2  15    8i 

6    24 

79 

2    14 

2    44'2    71 

2  10413    If  13    43 

3    8 

3  114 

4    2i 

4    5-.V 

4    845    23'5    94 

6    34 

802    14 

2    4s  2    7f 

2  11 

3    2i 

3    5g 

3    84 

3  llf 

4    3 

4    64 

4    9i 

5    3f|5  10 

6    4f 

81 

2   IS 

2    5    2    Si 

2114 

3    2s 

3    53 

3    94 

4    Oi 

4    34 

4    63 

410 

5    44  5  103 

6    5f 

822    24 

2    5|  2    8f 

21133    343    6s 

3    9§4    03 

4    4i 

4    74 

410S 

5    5i  5  Hi 

6    6| 

832    24 

2    5i2    9 

3    Of  3    3fl3    63 

3  10i!4    14 

4    43 

4    84 

4114 

5    6 

6    Of 

6    7i 

842    25 

2    6 

2    9| 

3    Oi 

3    4 

3    74 

3  lOf 

4    24 

4    54 

4    83 

5    04 

5    63 

6   14 

6    84 

85 

2    3 

2    6|  2    9f 

3    li 

3    443    73 

3  Hi 

4    2S 

4    64 

4    94 

5    03 

5    7s 

6    2S 

6    94 

862    3£ 

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3    If 

3    5il3    84 

3  113 

4    3i 

4    6S 

4104 

5    If 

5    846    3i 

6  104 

87 

2    3f 

2    74 

2  lOf 

3    2 

3    54 

3    9 

4    04 

4    33 

4    7f 

4103 

5    2i 

5    9i6    44 

611 

88|2    4 

2    74 

2  11 

3    24 

3    6 

3    94 

4    1 

4    44 

4    8 

4114 

5    3 

5  10 

6    5 

7    0 

892    4| 

2    73 

2  llf 

3    23 

3    64 

3  10 

4    14 

4    54 

4    8s 

5    Oi 

5    3i 

5  lOf 

6    53 

7    1 

90'2   4s 

2    Si 

2  Hi 

3    3i 

3    7 

3  104 

4    24 

4    5S 

4    9i  5    Of 

5    4i 

5  llf 

6    6S 

7    2 

91 

2    43 

2    84 

3    Oi 

3    33 

3    74 

3  11 

4    2| 

4    6i 

4    93 

5    14 

5    54 

6    Of 

6    7f 

7    23 

922    5i 

2    83 

3    Of 

3    4i 

3    73 

3  llf 

4    3i 

4    7 

4104 

5    24 

5    53 

6    1 

6    84 

7    33 

932    5| 

2    9i 

3    1 

3    4| 

3    8f 

4    04 

4    33 

4    74 

4  Hi  5    23 

5    6| 

6    2 

6    9| 

7    43 

942    53 

2    9| 

3    If 

3    54 

3    83)4    Of 

4    4i|4    84 

4  Hi  5    34 

5    7s 

6    2S 

6  10i 

7    5f 

952    6i 

2  10 

3    IS 

3    54 

3    9f|4    14 

4    434    8i 

5    045    4i 

5    8 

6    34 

6  114 

7    6S 

962    64 

2  lOf 

3    24 

3    6 

3    9S4    If 

4    54 

4    9f 

5    145    5 

5    8S 

6    4f 

7    0 

7    7f 

972    63 

2  10S 

3    2s 

3    64 

3  10i  4    24 

4    6 

410 

5    li  5    5f 

5    94 

6    5i 

7    03 

7    8| 

982    74 

2  11 

3    3 

3    63 

3  lOf 

4    2| 

4    64 

4  104 

5    2f 

5    6i 

5  104 

6    6 

7   li 

7    94 

992    74 

2  llf 

3    3| 

3    7t 

3  Hi 

4    3i 

4    74 

411 

5    3 

5    7 

5  11 

6    6f 

7  55 

7  104 

1002    73 

2  113 

3    3S 

3    7S 

3  IIS 

4    3S 

4    7S 

411f 

5    3f 

5    7s 

5  llf 

6    74 

7    3f 

7  114 

1012    8i 

3    04 

3    44 

3    8i 

4    Oi 

4    4i 

4    8i 

5    Oi 

5    4i 

5    8i 

6    Oi 

6    8f 

7    4| 

8    04 

1022    84 

3    04 

3    44 

3    Sf 

4    Of  4    4S4    835    1 

5    5 

5    9 

6    1 

6    94 

7    5i 

8    1| 

TABLE  of  the  Strength  of  the  Teeth  of  Cast  Iron  Wheels  at  a 
given  velocity. 


Pitch 
of  teeth 
in  inches. 

Thickness 
of  teeth 
in  inches. 

Breadth 
of  teeth 
in  inches. 

Strength  of  teeth  in  horse  power,  at 

3  feet  per 
second. 

4  feet  per 
secoud. 

6  feet  per 
second. 

8  feet  per 
second. 

3-99 

1-9 

7-6 

20-57 

2743 

41-14 

54-85 

3-78 

1-8 

7-2 

1749 

23-32 

34-98 

46-64 

3-57 

1-7 

6-8 

14-73 

19-65 

29-46 

39-28 

3-36 

1-6 

64 

12-28 

16-38 

24-56 

32-74 

3-15 

1-5 

6 

10-12 

13-50 

20-24 

26-98 

2-94 

14 

5-6 

8-22 

10-97 

1644 

21-92 

2-73 

1-3 

5-2 

6-58 

8-78 

13-16 

17-54 

2-52 

1-2 

4-8 

5-18 

6-91 

10-36 

13-81 

2-31 

1-1 

44 

3-99 

5-32 

7-98 

10-64 

2-1 

1-0 

4 

3-00 

4-00 

6-00 

8-00 

1-89 

•9 

3-6 

2-18 

2-91 

4-36 

5-81 

1-68 

•8 

3-2 

1-53 

2-04 

3-06 

3-08 

147 

•7 

2-8 

1-027 

1-37 

2-04 

2-72 

1-26 

•6 

24 

•64 

•86 

1-38 

1-84 

1-05 

•5 

2 

•375 

•50 

•75 

1-00 

438  THE   PRACTICAL   MODEL   CALCULATOR. 


ADDITIONAL  EXAMPLES  ON  THE  VELOCITY  OF  WHEELS, 
DRUMS,  PULLEYS,  ETC. 

IF  a  wheel  that  contains  75  teeth  makes  16  revolutions  per 
minute,  required  the  number  of  teeth  in  another  to  work  in  it,  and 
make  24  revolutions  in  the  same  time. 

75  x  16 

— 24 —  =  50  teeth- 

A  wheel,  64  inches  diameter,  and  making  42  revolutions  per 
minute,  is  to  give  motion  to  a  shaft  at  the  rate  of  77  revolutions 
in  the  same  time :  required  the  diameter  of  a  wheel  suitable  for 
that  purpose. 

64  x  42 
— == —  =  34-9  inches. 

Required  the  number  of  revolutions  per  minute  made  by  a  wheel 
,  or  pulley  20  inches  ^diameter,  when  driven  by  another  of  4  feet  di- 
ameter, and  making  46  revolutions  per  minute. 

48  X  46 

— OA —  =  110-4  revolutions. 

A  shaft,  at  the  rate  of  22  revolutions  per  minute,  is  to  give  mo- 
tion, by  a  pair  of  wheels,  to  another  shaft  at  the  rate  of  15J ;  the 
distance  of  the  shafts  from  centre  to  centre  is  45J  inches ;  the  di- 
ameters of  the  wheels  at  the  pitch  lines  are  required. 

22    4-  15-5  ~  -^'81  radius  of  the  driving  wheel. 

45*5  x    22 
And   00    4.  15.5  ~  26-69  radius  of  the  driven  wheel. 

Suppose  a  drum  to  make  20  revolutions  per  minute,  required  the 
diameter  of  another  to  make  58  revolutions  in  the  same  iime. 

58  -T-  20  =  2-9,  that  is,  their  diameters  must  be  as  2-9  to  1 ; 
thus,  if  the  one  making  20  revolutions  be  called  30  inches,  the 
other  will  be  30  -i-  2-9  =  10-345  inches  diameter. 

Required  the  diameter  of  a  pulley,  to  make  12J  revolutions  in 
the  same  time  as  one  of  32  inches  making  26. 

32  x  26 

—  =  66*56  inches  diameter. 

A  shaft,  at  the  rate  of  16  revolutions  per  minute,  is  to  give  mo- 
tion to  a  piece  of  machinery  at  the  rate  of  81  revolutions  in  the 
same  time ;  the  motion  is  to  be  communicated  by  means  of  two 
wheels  and  two  pulleys  with  an  intermediate  shaft;  the  driving 
wheel  contains  54  feet,  and  the  driving  pulley  is  25  inches  diameter ; 
required  the  number  of  teeth  in  the  other  wheel,  and  the  diameter 
of  the  other  pulley. 


VELOCITY   OF    WHEELS,    ETC. 


439 


v/81  X  16  =  36,  the  mean  velocity  between  16  and  81 ;  then, 
— og —  =  24  teeth;  and  — 01 —  =  ll'll  inches,  diameter  of 
pulley. 

Suppose  in  the  last  example  the  revolutions  of  one  of  the  wheels 
to  be  given,  the  number  of  teeth  in  both,  and  likewise  the  diameter 
of  each  pulley,  to  find  the  revolutions  of  the  last  pulley. 

— ^-j —  =  36,  velocity  of  the  intermediate  shaft ; 

36  x  25 
and     11.11     ^  81»  the  velocity  of  the  machine. 

TABLE  for  finding  the  radius  of  a  wheel  when  the  pitch  is  given,  or 
the  pitch  of  a  wheel  when  the  radius  is  given,  that  shall  contain 
from  10  to  150  teeth,  and  any  pitch  required. 


Number 
of  Teeth. 

Radius. 

Number 
of  Teeth. 

Radius. 

Number 
of  Teeth. 

Radius. 

Number 
of  Teeth. 

-  Radius. 

10 

1-618 

46 

7-327 

81 

12-895 

116 

18-464 

11 

1-774 

47 

7-486 

82 

13-054 

117 

18-623 

12 

1-932 

48 

7-645 

83 

13-213 

118 

18-782 

13 

2-089 

49 

7-804 

84 

13-370 

119 

18-941 

14 

2-247 

50 

7-963 

85 

13-531 

120 

19-101 

15 

2-405 

51 

8-122 

86 

13-690 

121' 

19-260 

16 

2-563 

52 

8-281 

87 

13-849 

122 

19-419 

17 

2-721 

53 

8-440 

88 

14-008 

123 

19-578 

18 

2-879 

54 

8-599 

89 

14-168 

124 

19-737 

19 

3-038 

55 

8-758 

90 

14-327 

125 

19-896 

20 

3-196 

56 

8-917 

91 

14-486 

126 

20-055 

21 

3-355 

57 

9-076 

92 

14-645 

127 

20-214 

22 

3-513 

58 

9-235 

93 

14-804 

128 

20-374 

23 

3-672 

59 

9-394 

94 

14-963 

129 

20-533 

24 

3-830 

60 

9-553 

95 

15-122 

130 

20-692 

25 

3-989 

61 

9-712 

96 

15-281 

131 

20-851 

26 

4-148 

62 

9-872 

97 

15-440 

132 

21-010 

27 

4-307 

63 

10-031 

98 

15-600 

133 

21-169 

28 

4-465 

64 

10-190 

99 

15-759 

134 

21-328 

29 

4-624 

65 

10-349 

100 

15-918 

135 

21-488 

30 

4-788 

66 

10-508 

101 

16-077 

136 

21-647 

31 

4-942 

'  67 

10-667 

102 

16-236 

137 

21-806 

32 

5-101 

68 

10-826 

103 

16-395 

138 

21-965 

33 

5-260 

69 

10-985 

104 

16-554 

139 

22-124 

34 

5-419 

70 

11-144 

105 

16-713 

140 

22-283 

35 

5-578 

71 

11-303 

106 

16-878 

141 

22-442 

36 

5-737 

72 

11-463 

107 

17-032 

142 

22-602 

37 

5-896 

73 

11-622 

108 

17-191 

143 

22-761 

38 

6-055 

74 

11-781 

109 

17-350 

144 

22-920 

39 

6-214 

75 

11-940 

110 

17-509 

145 

23-079 

40 

6-373 

76 

12-099 

111 

17-668 

146 

23-238 

41 

6-532 

77 

12-258 

112 

17-827 

147 

23-397 

42 

6-691 

78 

12-417 

113 

17-987 

148 

23-556 

43 

6-850 

79 

12-576 

114 

18-146 

149 

23-716 

44 

7-009 

80 

12-735 

115 

18-305 

150 

23-875 

45 

7-168 

RULE. — Multiply  the  radius  in  the  table  by  the  pitch  given,  and 
the  product  will  be  the  radius  of  the  wheel  required. 


440 


THE  PRACTICAL  MODEL  CALCULATOR. 


Or,  divide  the  radius  of  the  wheel  by  the  radius  in  the  table, 
and  the  quotient  will  be  the  pitch  of  the  wheel  required. 

Required  the  radius  of  a  wheel  to  contain  64  teeth,  of  3  inch 
pitch. 

10-19  x  3  =  30-57  inches. 

What  is  the  pitch  of  a  wheel  to  contain  80  teeth,  when  the  radius 
is  25-47  inches  ? 

25-47  -f-  12-735  =  2  inch  pitch. 

Or.  set  off  upon  a  straight  line  AB  seven  times  the  pitch  AC 
given  ;  divide  that,  or  another  exactly  the  same  length,  into  eleven 
equal  parts ;  call  each  of  those  divisions  four,  or  each  of  those  di- 
visions will  be  equal  to  four  teeth  upon  the  radius.  If  a  circle  be 
made  with  any  number  (20)  of  these  equal  parts  as  radius,  AC  the 
pitch  will  go  that  number  (20)  of  times  round  the  circle. 


\ 


Were  it  required  to  find  the  diameter  of  a  wheel  to  contain  17 
teeth,  the  construction  would  be  as  follows : — 


A 

1     < 

-1 

2 

3 

4 

5 

6 

7 

B 

1 

1 

! 

1 

4 

5 

6 

7 

8 

9 

i 

0 

11 

a 

i  (  i 

1 

1 

1 

1 

i   1   1 

j? 

1   1   1 

1  1 

1    1 

1   :  I 

I 

1 

I   i 

g 

Thus,  4  divisions  and  J  of  another  equal  the  radius  of  the  wheel, 
that  is  a±b^  —  a  b,  and  Aa  Ca  =  AC. 


VELOCITY   OP  WHEELS,   ETC. 


441 


Regular  approved  proportions  for  wheels  with  flat  arms  in  the 
middle  of  the  ring,  and  ribs  or  feathers  on  each  side. — The  length 
of  the  teeth  =  f  the  pitch,  besides  clearance,  or  f  the  pitch,  clear- 
ance included. 

Thickness  of  the  teeth f  the  pitch. 

Breadth  on  the  face 2£       — 

Edge  of  the  rim f       — 

Rib  projecting  inside  the  rim |       — 

Thickness  of  the  flat  arms ».  f       — 

Breadth  of  the  arms  at  the  points  =  2  teeth  and  J  the  pitch, 
getting  broader  towards  the  centre  of  the  wheel  in  the  proportion 
of  J  inch  to  every  foot  in  length. 

Thickness  of  the  ribs,  or  feathers,  J  the  pitch. 

Thickness  of  metal  round  the  eye,  or  centre,  f  the  pitch. 

Wheels  made  with  plain  arms,  the  teeth  are  in  the  same  propor- 
tion as  above  ;  the  ring  and  the  arms  are  each  equal  to  one  cog  or 
tooth  in  thickness,  and  the  metal  round  the  eye  same  as  above,  in 
feathered  wheels. 

These  proportions  differ,  though  slightly,  in  different  works  and 
in  different  localities ;  but  they  are  the  most  commonly  employed, 
and  are  besides  the  most  consistent  with  good  and  accurate  work- 
manship. For  the 
sake  of  more  easy 
reference,  we  col- 
lect them  into  a 
table,  which  the 
annexed  diagram 
will  serve  fully  to 
explain.  They 
stand  thus : — 

a  t>  =  Pitch  of  teeth  =  1  pitch. 

mw,  =  Depth  to  pitch  line,  PP,  =  $  — . 

n  s  +  n  m  —  Working  depth  of  tooth,  =  ^  — . 

C  b  — -  n  s  =  Bottom  clearance,  =  ^  — . 

fh  =  Whole  depth  to  root,        =  ^  — . 

p  q  —  Thickness  of  tooth,  =  T5T  — . 

rp  =  Width  of  space,  =  £  — . 

The  use  of  the  following  table  is  very  evident,  and  the  manner 
of  applying  it  may  be  rendered  still  more  obvious  by  the  following 
examples : — 

*  =  3-1416. 

1.  Given  a  wheel  of  88  teeth,  2J  inch  pitch,  to  find  the  diameter 
of  the  pitch  circle.     Here  the  tabular  number  in  the  second  column 
answering  to  the  given  pitch  is  -7958,  which  multiplied  by  88  gives 
70'03  for  the  diameter  required. 

2.  Given  a  wheel  of  5  feet  (60  inches)  diameter,  2}  inch  pitch, 
to  find  the  number  of  teeth.     Here  the  factor  in  the  third  column 


442 


THE  PRACTICAL  MODEL  CALCULATOR. 


corresponding  to  the  given  pitch  is 
1-1333,  which  multiplied  by  60  gives 
68  for  the  number  of  teeth. 

It  may,  however,  so  happen  that  the 
answer  found  in  this  manner  contains  a 
fraction — which  being  inadmissible  by 
the  nature  of  the  question,  it  becomes 
necessary  to  alter  slightly  the  diameter 
of  the  pitch  circle.  This  is  readily  ac- 
complished by  taking  the  nearest  whole 
number  to  the  answer  found,  and  find- 
ing the  modified  diameter  by  means  of 
the  second  column.  The  following  case 
will  fully  explain  what  is  meant  : 

3.  Given  &>  wheel  33  inches  diameter, 
1}  inch  pitch,  to  find  the  number  of 
teeth.  The  corresponding  factor  is 
1-7952,  which  multiplied  by  33  gives 
59-242  for  the  number  of  teeth,  that  is, 
59J  teeth  nearly.  Now,  59  would  here 
be  the  nearest  whole  number ;  but  as  a 
wheel  of  60  teeth  may  be  preferred  for 
convenience  of  calculation  of  speeds,  we 
may  adopt  that  number  and  find  the  di- 
ameter corresponding.  The  factor  in 
the  second  column  answering  to  If  pitch 
is  -55T,  and  this  multiplied  by  60  gives 
33-4  inches  as  the  diameter  which  the 
wheel  ought  to  have.v 

RULE. — To  find  the  power  that  a  cast  iron  wheel  is  capable  of 
transmitting  at  any  given  velocity. — Multiply  the  breadth  of  the 
teeth,  or  face  of  the  wheel,  in  inches,  by  the  square  of  the  thick- 
ness of  one  tooth,  and  divide  the  product  by  the  length  of  the  teeth, 
the  quotient  is  the  strength  in  horse  power  at  a  velocity  of  136 
feet  per  minute. 

Required  the  power  that  a  wheel  of  the  following  dimensions 
ought  to  transmit  with  safety,  namely, 

Breadth  of  teeth 7J  inches, 

Thickness , 1-4 

And  length 2 

1-4*  =  1-96,  and  7-5  x  1-96 

2 =  7-35  horse  power. 

The  strength  at  any  other  velocity  is  found  by  multiplying  the 
power  so  obtained  by  any  other  required  velocity,  and  by  -0044, 
the  quotient  is  the  power  at  that  velocity. 

Suppose  the  wheel  as  above,  at  a  velocity  of  320  feet  per  minute. 
7-35  X  320  X  -0044  =  10-3488  horse  power. 


P 
D=:-XN 

N  =  -  X  D' 

7T 

P 

Pitch  in 
inches  and 

RDLE.-TO    find 
the  diameter  In 

UULS.-TO    Kn-.l 
the  number  of 

parts  of  au 
inch. 

inches,    multi- 
ply the  number 
of  teeth  by  the 

teeth,  multiplr 
the  given  dia- 
meter in  inctu'3 

tabular    num- 

by the  tabular 

ber    answering 

number        an- 

to    the    given 

swering  to  the' 

pitch. 

given  pitch. 

Values  of  P 

Values  of  - 

Values  of  - 

7T 

1' 

6 

1-9095 

•5236 

5 

1-5915 

•6283 

^ 

1-4270 

•6981 

4 

1-2732 

•7854 

^ 

1-1141 

•8976 

3 

•9547 

l-0i72  j 

93. 

•8754 

1-1333 

2f 

•7958 

1-2566 

21 

•7135 

1-3963 

2 

•6366 

1-5708 

if 

•5937 

1-6755 

1! 

•5570 

1-7952 

If 

•5141 

1-9264 

i!: 

•4774 

2-0944 

if 

•4377 

2-2848 

1; 

•3979 

2-5132 

lr 

•3568 

2-7926 

1 

•3183 

3-1416 

1 

•2785 

3-5904 

•2387 

4-1888 

-| 

•1989 

5-0266 

1 

•1592 

6-2832 

! 

-•1194 

8-3776 

I 

•0796 

12-5664 

MAXIMUM   VELOCITY   AND    POWER   OF  WATER   WHEELS.      443 


ON  THE  MAXIMUM  VELOCITY  AND  POWER  OF  WATER 

WHEELS. 

OP   UNDERSHOT  WHEELS. 

THE  term  "undershot"  is  applied  to  a  wheel  when  the  water 
strikes  at,  or  below,  the  centre  ;  and  the  greatest  effect  is  produced 
when  the  periphery  of  the  wheels  moves  with  a  velocity  of  '57  that 
of  the  water  ;  hence,  to  find  the  velocity  of  the  water,  multiply  the 
square  root  or  the  perpendicular  height  of  the  fall  in  feet  by  8, 
and  the  product  is  the  velocity  in  feet  per  second. 

Required  the  maximum  velocity  of  an  undershot  wheel,  when 
propelled  by  a  fall  of  water  6  feet  in  height. 


=  245  x  8  =  19-6  feet,  velocity  of  water. 
And  19*6  x  '57  =  11*17  feet  per  second  for  the  wheel. 

OP  BREAST  AND    OVERSHOT   WHEELS. 

Wheels  that  have  the  water  applied  between  the  centre  and  the 
vertex  are  styled  breast  wheels,  and  overshot  when  the  water  is 
brought  over  the  wheel  and  laid  on  the  opposite  side  ;  however,  in 
either  case  the  maximum  velocity  is  f  that  of  the  water  ;  hence, 
to  find  the  head  of  water  proper  for  a  wheel  at  any  velocity,  say  : 

As  the  square  of  16-083,  or  258-67,  is  to  4,  so  is  the  square  of 
the  velocity  of  the  wheel  in  feet  per  second  to  the  head  of  water 
required.  By  head  is  understood  the  distance  between  the  aper- 
ture of  the  sluice  and  where  the  water  strikes  upon  the  wheel. 

Required  the  head  of  water  necessary  for  a  wheel  of  24  feet 
diameter,  moving  with  a  velocity  of  5  feet  per  second. 

5x3 

—  2  —  =  7*5  feet,  velocity  of  the  water. 

And  258-67  :  4  :  :  7'52  :  -87  feet,  head  of  water  required. 

But  one-tenth  of  a  foot  of  head  must  be  added  for  every  foot 
of  increase  in  the  diameter  of  the  wheel,  from  15  to  20  feet,  and 
•05  more  for  every  foot  of  increase  from  20  to  30  feet,  'commencing 
with  five-tenths  for  a  15  feet  wheel. 

This  additional  head  is  intended  to  compensate  for  the  friction 
of  water  in  the  aperture  of  the  sluice  to  keep  the  velocity  as  3  to  2 
of  the  wheel  ;  thus,  in  place  of  -87  feet  head  for  a  24  feet  wheel, 
it  will  be  -87  -f  1-2  =  2-07  feet  head  of  water. 

If  the  water  flow  from  under  the  sluice,  multiply  the  square 
root  of  the  depth  in  feet  by  5-4,  and  by  the  area  of  the  orifice  also 
in  feet,  and  the  product  is  the  quantity  discharged  in  cubic  feet 
per  second. 

Again,  if  the  water  flow  over  the  sluice,  multiply  the  square 
root  of  the  depth  in  feet  by  5-4,  and  f  of  the  product  multiplied 


444  THE   PRACTICAL   MODEL   CALCULATOR. 

by  the  length  and  depth,  also  in  feet,  gives  the  number  of  cubic 
feet  discharged  per  second  nearly. 

Required  the  number  of  cubic  feet  per  second  that  will  issue 
from  the  orifice  of  a  sluice  5  feet  long,  9  inches  wide,  and  4  feet 
from  the  surface  of  the  water. 


2  X  5-4  =  10-8  feet  velocity. 
And  5  X  -75  X  10-8  =  40-5  cubic  feet  per  second. 

What  quantity  of  water  per  second  will  be  expended  over  a  wear, 
dam,  or  sluice,  whose  length  is  10  feet,  and  depth  6  inches  ? 

1*20744  x  2 
'  v/-5  =  -2236  x  5-4  =  -     —  g—     -  =  -80496  feet  velocity. 

Then  10  X  -5  =  5  feet,  and  -80496  x  5  =  4-0248  cubic  feet 
per  second  nearly. 

In  estimating  the  power  of  water  wheels,  half  the  head  must  be 
added  to  the  whole  fall,  because  1  foot  of  fall  is  equal  to  2  feet  of 
head  ;  call  this  the  effective  perpendicular  descent  ;  multiply  the 
weight  of  the  water  per  second  by  the  effective  perpendicular  de- 
scent and  by  60  ;  divide  the  product  by  33,000,  and  the  quotient 
is  the  effect  expressed  in  horse  power. 

Given  16  cubic  feet  of  water  per  second,  to  be  applied  to  an  under- 
shot wheel,  the  head  being  12  feet  ;  required  the  power  produced. 

6  x  16  x  62-5  x  60 
12  -r-  2  =  6  and  -      —  33QQQ  —      ~  ^  •^'"  norse  power  nearly. 

Given  1(5  cubic  feet  of  water  per  second,  to  be  applied  to  a  high 
breast  or  an  overshot  wheel,  with  2  feet  head  and  10  feet  fall  ; 
required  the  power. 

1  +  10  x  16  x  62-5  x  60 
2  -r-  2  =  1  and  —        —  33QQO  --  =  20  horse  power. 

Only  about  two-thirds  of  the  above  results  can  be  taken  as  real 
communicative  power  to  machinery. 

OF  THE   CIRCLE   OF  GYRATION   IN   WATER   WHEELS. 

The  centre  or  circle  of  gyration  is  that  point  in  a  revolving  body 
into  which,  if  the  whole  quantity  of  matter  were  collected,  the 
same  moving  force  would  generate  the  same  angular  velocity, 
which  renders  it  of  the  utmost  importance  in  the  erection  of  water 
wheels,  and  the  motion  ought  always  to  be  communicated  from  that 
point  when  it  is  possible. 

RULE.  —  To  find  the  circle  of  gyration.  —  Add  into  one  sum  twice 
the  weight  of  the  shrouding,  buckets,  &c.,  multiplied  by  the  square 
of  the  radius,  f  of  the  weight  of  the  arms,  multiplied  by  the  square 
of  the  radius,  and  the  weight  of  the  water  multiplied  by  the  square 
of  the  radius  also  ;  divide  the  sum  by  twice  the  weight  of  the 
shrouding,  arms,  &c.,  added  to  the  weight  of  the  water,  and  the 
square  root  of  the  quotient  is  the  distance  of  the  circle  of  gyra- 
tion from  the  centre  of  suspension  nearly. 


MAXIMUM   VELOCITY  AND   POWER   OF   WATER  WHEELS.      445 


Required  the  distance  of  the  centre  of  gyration  from  the  centre 
of  suspension  in  a  water  wheel  22  feet  diameter,  shrouding,  buckets, 
&c.  =  18  tons,  arms  =  12  tons,  and  water  ==  10  tons. 

22  -T-  2  =  11  and  II2  =    121 

Then,  18  x  2  =  36  x  121  =  4356 

f  of  12  =  8  x  121  =  968 

water  =  10  x  121   =  1210 

6534 


And  18  +  12  X  2  =  60  +  10  =  70;  hence, 

9 '6  feet  from  the  centre  of  suspension  nearly. 


T 


TABLE  of  Angles  for  Windmill  Sails. 


Number. 

Angle  with  the  Plane  of  Motion. 

1 

18° 

24° 

2 

19 

21 

3 

18 

18 

4 

16 

14 

5 

12A 

9 

6 

7 

3  extremity. 

The  radius  is  supposed  to  be  divided  into  six  equal  parts,  and  J 
from  the  centre  is  called  1,  the  extremity  being  denoted  by  6. 

The  first  column  contains  the  angles  according  to  an  old  custom ; 
but  experience  has  taught  us  that  the  angles  in  the  second  column 
are  preferable. 


THE    VELOCITY  OF   THRESHING   MACHINES, 
IRON,    ETC. 


MILLSTONES,   BORING 


The  drum  or  beaters  of  a  threshing  machine  ought  to  move  with 
a  velocity  of  about  3000  feet  per  minute ;  hence,  divide  11460  by 
the  diameter  of  the  drum  in  inches ;  or  955  by  the  diameter  of  the 
drum  in  feet;  and  the  quotient  is  the  number  of  revolutions 
required  per  minute.  And  the  feeding  rollers  must  make  half 
the  revolutions  of  the  drum,  when  their  diameters  are  about 
3J  inches. 

If  the  machine  is  driven  by  horses,  their  velocity  ought  to  be 
from  2J  to  3  times  round  a  24  feet  ring  per  minute. 

Divide  500  by  the  diameter  of  a  millstone,  in  feet,  or  6000  by 
the  diameter  in  inches,  and  the  quotient  is  the  number  of  revolu- 
tions required  per  minute. 

In  boring  cast  iron  the  cutters  ought  to  have  a  velocity  of  about 
108  inches  per  minute,  or  divide  36  by  the  diameter  in  inches,  the 
quotient  is  the  number  of  revolutions  of  the  boring  head  per  minute. 
And  divide  100  by  the  diameter  in  inches,  the  quotient  is  the  num- 
ber of  revolutions  per  minute,  for  turning  wrought  iron  in  general, 
and  about  half  that  velocity  for  cast  iron. 


446  THE    PRACTICAL   MODEL    CALCULATOR. 


OF  PUMPS  AND  PUMPING  ENGINES, 

PUMPS  are  chiefly  designated  by  the  names  of  lifting  and  force 
pumps  ;  lifting  pumps  are  applied  to  wells,  &c.,  where  the  height 
of  the  bucket,  from  the  surface  of  the  water,  must  not  exceed  33  feet  ; 
this  being  nearly  equal  to  the  pressure  of  the  atmosphere,  or  the 
height  to  which  water  would  be  forced  up  into  a  vacuum  by  the 
pressure  of  the  atmosphere.  Force  pumps  are  applicable  on  all 
other  occasions,  as  raising  water  to  any  required  height,  supplying 
boilers  against  the  force  of  the  steam,  hydrostatic  presses,  &c. 

The  power  required  to  raise  water  to  any  height  is  as  the  weight 
and  velocity  of  the  water  with  an  addition  of  about  f  of  the  whole 
power  for  friction  ;  hence  the 

RULE.  —  Multiply  the  perpendicular  height  of  the  water,  in  feet, 
by  the  velocity,  also  in  feet,  and  by  the  square  of  the  pump's 
diameter  in  inches,  and  again  by  -341  ;  (this  being  the  weight  of 
.  a  column  of  water  1  inch  diameter,  and  12  inches  high,  in  Ibs. 
avoirdupois  ;)  divide  the  product  by  33,000,  and  f  of  the  quotient 
added  to  the  whole  quotient  will  be  the  number  of  horse  power 
required. 

Required  the  power  necessary  to  overcome  the  resistance  and 
friction  of  a  column  of  water  4  inches  diameter,  60  feet  high,  and 
flowing  with  a  velocity  of  130  feet  per  minute. 

60  x  130  x  42  x  -341      1-8 

—  =  --  =  *26  +  1*3  =  156  horse  power  nearly. 


Hot  liquor  pumps,  or  pumps  to  be  employed  in  raising  any  fluid 
where  steam  is  generated,  require  to  be  placed  in  the  fluid,  or  as 
low  as  the  bottom  of  it,  on  account  of  the  steam  filling  the  pipes, 
and  acting  as  a  counterpoise  to  the  atmosphere  ;  and  the  diameter  of 
the  pipes  to  and  from  a  pump  ought  not  to  be  less  than  f  of  the 
pump's  diameter. 

RULE.  —  The  diameter  of  a  pump  and  velocity  of  the  water  given, 
to  find  the  quantity  discharged  in  gallons,  or  cubic  feet,  in  any 
given  time.  —  Multiply  the  velocity  of  the  water,  in  feet  per  minute, 
by  the  square  of  the  pump's  diameter  in  inches,  and  by  -041  for 
gallons,  or  -000545<f  for  cubic  feet,  and  the  product  will  be  the 
number  of  gallons,  or  cubic  feet,  discharged  in  the  given  time 
nearly. 

What  is  the  number  of  gallons  of  water  discharged  per  hour  by 
a  pump  4  inches  diameter,  the  water  flowing  at  the  rate  of  130 
feet  per  minute  ? 

130  x  60  =  7800  feet  per  hour. 
And,  7800  X  42  X  -041  =  5116-8  gallons. 

RULE  1.  —  The  length  of  stroke  and  number  of  strokes  given,  to 
find  the  diameter  of  a  pump,  and  number  of  horse  power  that  will 
discharge  a  given  quantity  of  water  in  a  given  time.  —  Multiply  the 


OF   PUMPS   AND   PUMPING   ENGINES.  447 

number  of  cubic  feet  by  2201,  and  divide  the  product  by  the  ve- 
locity of  the  water,  in  inches,  and  the  square  root  of  the  quotient 
will  be  the  pump's  diameter,  in  inches. 

2.  Multiply  the  number  of  cubic  feet  by  62-5,  and  by  the  per- 
pendicular height  of  the  water  in  feet,  divide  the  product  by  33,000, 
then  will  f  of  the  quotient,  added  to  the  whole  quotient,  be  the 
number  of  horse  power  required. 

Required  the  diameter  of  a  pump,  and  number  of  horse  power, 
capable  of  filling  a  cistern  20  feet  long,  12  feet  wide,  and  6J  feet 
deep,  in  45  minutes,  whose  perpendicular  height  is  53  feet ;  the 
pump  to  have  an  effective  stroke  of  26  inches,  and  make  30  strokes 
per  minute. 

20  X  12  x  6-5  =  1560  cubic  feet,  and 

1560 

— — -  =  34-66  cubic  feet  per  minute. 

Then,  34-66  x  2201 

,na OQ~~  ~  9-89  inches  diameter  of  pump. 

And  34-66  x  62-5  x  53        3-48 

33000          -  =  -5-  =  '69  +  3-48  =  4-17  horse 

power. 

RULE. —  To  find  the  time  a  cistern  will  take  in  filling,  when  a 
known  quantity  of  water  is  going  in,  and  a  known  portion  of  that 
water  is  going  out,  in  a  given  time. — Divide  the  content  of  the  cis- 
tern, in  gallons,  by  the  difference  of  the  quantity  going  in,  and 
the  quantity  going  out,  and  the  quotient  is  the  time  in  hours  and 
parts  that  the  cistern  will  take  in  filling. 

If  30  gallons  per  hour  run  in  and  22J  gallons  per  hour  run  out 
of  a  cistern  capable  of  containing  200  gallons,  in  what  time  will 
the  cistern  be  filled  ? 

30  -  22-5  =  7-5,  and  200  -4-  7-5  =  26-666,  or  26  hours  and 
40  minutes. 

To  find  the  time  a  vessel  will  take  in  emptying  itself  of  water. — 
Mr.  O'Neill  ascertained,  from  very  accurate  experiments,  that  a 
vessel,  3-166  feet  long  and  2-705  inches  diameter,  would  empty  it- 
self in  3  minutes  and  16  seconds,  through  an  orifice  in  the  bottom, 
whose  area  is  '0141  inches ;  and  another  6-458  feet  long,  the  dia- 
meter and  orifice,  as  before,  would  do  the  same  in  4  minutes  and  40 
seconds ;  hence,  from  these  experiments,  a  rule  is  obtained,  namely, 

Multiply  the  square  root  of  the  depth  in  feet  by  the  area  of  the 
falling  surface  in  inches,  divide  the  product  by  the  area  of  the  ori- 
fice, multiplied  by  3*7,  and  the  quotient  is  the  time  required  in 
seconds,  nearly. 

How  long  will  it  require  to  empty  a  vessel  of  water,  9  feet  high, 
and  20  inches  diameter,  through  a  hole  f  inch  in  diameter  ? 
%/9  =  3,  the  square  root  of  the  depth, 
314-16  inches,  area  of  the  falling  surface, 
•4417  inches,  area  of  the  orifice  ; 


448          THE  PRACTICAL  MODEL  CALCULATOR. 

Then,  314-16  x  3 

•4417  X  3-7  ~   •'."  seconc*s>  or  ^  nimutes  and  36  seconds. 

On  the  pressure  of  fluids.  —  The  side  of  any  vessel  containing  a 
fluid  sustains  a  pressure  equal  to  the  area  of  the  side,  multiplied  by 
half  the  depth  ;  thus, 

Suppose  each  side  of  a  vessel  to  be  12  feet  long  and  5  feet  deep, 
when  filled  with  water,  what  pressure  is  upon  each  side  ? 

12    x    5  =  60  feet,  the  area  of  the  side, 
2-5    feet  =  half  the  depth,  and   . 
62*5  Ibs.  =  the  weight  of  a  cubic  foot  of  water. 
Then,  60  x  2-5  x  62-5  =  9375  Ibs. 

RULE.  —  To  find  the  iveight  that  a  given  power  can  raise  by  a 
hydrostatic  press.  —  Multiply  the  square  of  the  diameter  of  the  ram 
in  inches  by  the  power  applied  in  Ibs.,  and  by  the  effective  leverage 
of  the  pump-handle  ;  divide  the  product  by  the  square  of  the  pump's 
diameter,  also  in  inches,  and  the  quotient  is  the  weight  that  the 
power  is  equal  to. 

What  weight  will  a  power  of  50  Ibs.  raise  by  means  of  a  hydro- 
static press,  whose  ram  is  7  inches  diameter,  pump  J,  and  the  ef- 
fective leverage  of  the  pump-handle  being  as  6  to  1  ? 

72  X  50  x  6 

=  19200  Ibs.,  or  8  tons  11  cwt. 


In  the  following  rules  for  pumping  engines  the  boiler  is  supposed 
to  be  loaded  with  about  2J  Ibs.  per  square  inch,  and  the  barometer 
attached  to  the  condenser  indicating  26  inches  on  an  average,  or 
13  Ibs.,  ==  15J  Ibs.,  from  which  deduct  J  for  friction,  leaves  a  pres- 
sure of  10  Ibs.  nearly  upon  each  square  inch  of  the  piston. 

RULE.  —  To  find  the  diameter  of  a  cylinder  to  work  a  pump  of  a 
given  diameter  for  a  given  depth.  —  Multiply  the  square  of  the 
pump's  diameter  in  inches  by  J  of  the  depth  of  the  pit  in  fathoms, 
and  the  square  root  of  the  product  will  be  the  cylinder's  diameter 
in  inches. 

Required  the  diameter  of  a  cylinder  to  work  a  pump  12  inches 
diameter  and  27  fathoms  deep. 

N/(122  x  9)  =  36  inches  diameter. 

RULE.  —  To  find  the  diameter  of  a  pump,  that  a  cylinder  of  a  given 
diameter  can  work  at  a  given  depth.  —  Divide  three  times  the  square 
of  the  cylinder's  diameter  in  inches  by  the  depth  of  the  pit  in  fa- 
thoms, and  the  square  root  of  the  quotient  will  be  the  pump's  di- 
ameter in  inches. 

What  diameter  of  a  pump  will  a  36-inch  cylinder  be  capable  of 
working  27  fathoms  deep  ? 


j 


362x3       -.o-     i.      j'- 

12  inches  diameter. 


w 


27 

RULE. —  To  find  the  depth  from  which  a  pump  of  a  given  diameter 
ill  work  by  means  of  a  cylinder  of  a  given  diameter. — Divide  three 


OF   PUMPS    AND    PUMPING    ENGINES.  449 

times  the  square  of  the  cylinder's  diameter  in  inches  by  the  square 
of  the  pump's  diameter  also  in  inches,  and  the  quotient  will  be  the 
depth  of  the  pit  in  fathoms. 

Required  the  depth  that  a  cylinder  of  36  inches  diameter  will 
work  a  pump  of  12  inches  diameter. 


x  3 

=  27  fathoms. 

An  inelastic  body  of  30  Ibs.  weight,  moves  with  a  3  feet  velo 
city,  and  is  struck  by  another  inelastic  body  having  a  7  feet  velo- 
city, the  two  will  then  proceed,  after  the  blow,  with  the  velocity 

50  x  7  +  30  x  3       350  +  90       44       11 

v  —  £n    i    on =  07; =  "cr  =  "9-  =  5J-  feet. 

oO  +  60  oU          -.      o  A 

To  cause  a  body  of  120  Ibs.  weight  to  pass  from  a  velocity  c2  = 
1 J  feet  into  a  2  feet  velocity  v,  it  is  struck  by  a  heavy  body  of  50 
Ibs.,  what  velocity  will  the  body  acquire  ?  Here 

(t,  -  e,)  M3  (2  -  1-5)  x  120  6 

<?!    =      V     +     ~      T^r ~      =     4     +     '  FTj "      —     ^     +     F    =     O*^J 

feet. 

Two  perfectly  elastic  spheres,  the  one  of  10  Ibs.  the  other  of  16 
Ibs.  weight,  impinge  with  the  velocities  12  and  6  feet  against  each 
other,  what  will  be  their  velocities  after  impact  ?  Here  Mt  =  10 
and  c^  =  12  feet,  but  Ma  =  16  and  cz  =  —  6  feet,  hence  the  loss 
of  velocity  of  the  first  body  will  be 

2  x  16  (12  +  6)      2  x  16  x  18 

c±  -  v,  =  -     1ft  \   1ft — '  = H~ =  22-154  feet ;  and 

1  1U  +  lo  zb 

9    ^    1  Q    vx    -|  Q 

the  gain  in  velocity  of  the  other,  va  —  cz  =  -  — ^g —    *  ==  13*846 

feet.  Erom  this  the  first  body  after  impact  will  recoil  with  the  ve- 
locity vt  =  12  —  22-154  =  —  10-154  feet ;  and  the  other  with 
that  of  —  6  +  13-846  =  7,846  feet.  Moreover,  the  measure  of 
via  viva  of  the  two  bodies  after  impact  =  M^2  +  M3va2  =  10  X 
10-1542  +  16  x  7-8462  =  1031  +  985  =  2016,  as  likewise  of  that 
before  impact,  namely:  M^2  +  Ma<?22  =  10  x  122  +  16  X  62  = 
1440  +  576  =  2016.  Were  these  bodies  inelastic,  the  first  would 

G     ~~~   V 

only  lose  in  velocity  -— o — *  =  11-077  feet,  and  the  other  gain 

jj      — .    /> 

^~2 — -  =  6-923  feet ;  the  first  would  still  retain,  after  impact,  the 

velocity  12  —  11-077  =  0-923  feet,  and  the  second  take  up  the 
velocity  —  6  +  6-923  =  0-923,  and  the  loss  of  mechanical  effect 
would  be  (2016  -  (10  +  16)  0-9232)  -*•  2^  =  (2016  -  2-22)  x 
0-0155  =  29-35  ft.  Ibs. 

29 


450 


CENTRIPETAL  AND  CENTRIFUGAL  FORCE. 

1.  WHAT  is  the  centrifugal  force  of  a  body  weighing  20  Ibs. 
that  describes  a  circle  of  10  feet  radius  200  times  in  a  minute  ? 

•000331  X  2002  x  20  X  10  =  2648  Ibs.,  the  centrifugal  force. 
•00331  is  a  constant  number. 

It  is  a  well  established  fact  that  the  centrifugal  force  is  to  the 
weight  of  the  body  as  double  the  height  due  to  the  velocity  is  to 
the  radius  of  revolution.  Hence,  this  question  may  be  thus  solved  : 

20  x  3-1416  =  62-832,  the  circumference  of  the  circle  of  10 
feet  radius. 

62-832  X  200  =  12&66-4  feet,  the  space  passed  over  by  the 
weight  in  one  minute. 

—  —  —  =  209-44  feet,  the  space  described  in  a  second,  which 

is  called  the  velocity. 

(209-44Y2 

v   6,  ,    •-  =  681-136  feet,  the  height  due  to  the  velocity. 

If  F  be  the  ^centrifugal  force  — 

F  :  20  :  :  1362-272  :  10. 

136^*272  x  20 
...  F  =  -        1Q        -  =  2724-544  Ibs.    The  former  rule  gives 

2648  Ibs. 

2.  What  is  the  centrifugal  force  at  the  equator  on  a  body  weigh- 
ing 300  Ibs.,  supposing  the  radius  of  the  earth  =  21000000  feet, 
and  the  time  of  rotation  =  86400"  =  24  hours? 


F  =  1-224  x  8°°  =  1-03298  lb,,  or  one  pound 


very  nearly.     1-224  is  a  constant  multiplier. 

3-1416   x  21000000  =  65973600  feet,  J  the  circumference 
of  the  earth  at  the  equator. 

-  00400  --  =  1527-16  feet>  tne  velocity  of  the  weight 
each  second. 

H-527'16)2 

v    64>4   ;   =  36214-56,  the  height  due  to  the  velocity. 

F  :  300  :  :  72429-12  :  21000000. 
72429-12  x  300 


21000000  nearly,  as  by  the  former  ap- 

proximate method. 

3.  If  a  body  weighing  100  Ibs.  describe  a  circle  of  10  feet  radius 
300  times  a  minute,  what  is  the  diameter  of  a  cast  iron  cylindrical 


CENTRIPETAL   AND   CENTRIFUGAL   FORCE. 


451 


rod,  connecting  the  body  with  the  axis,  that  will  safely  support  this 
weight  ?     The  centrifugal  force  will  be, 

•000331  x  3002  x  100  x  10  =  29790  Ibs. 

From  the  strength  of  materials,  page  281,  we  find  that  the  ulti- 
mate cohesive  strength  for  each  circular  inch  of  cross  sectional  area 
is  14652  Ibs.  ;  but  one-third  of  this  weight,  or  4884  Ibs.,  can  only 
be  applied  with  safety.  l 

/29790 

=  2-46982  inches,  the  diameter  of  the  cylindrical  rod. 


4.  The  dimensions,  the  density,  and  strength  of  a  millstone 
ABDE  are  given  ;  it  is  required  to  find  the  angular  velocity  v,  in 
consequence  of  which  rupture  will  take  place  on  account  of  the 
centrifugal  force. 


If  we  put  the  radius  of  the  millstone  =  rt  —  24  inches  = 
CG;  the  radius  =  CK  of  its  eye  =  r2  =  4  inches;  the  height 
PQ  =  GH  =  I  =  12  inches ;  the  density  =  t  =  2500  =  specific 
gravity  of  the  millstone;  and  the  modulus  of  strength  =  K  = 
750  Ibs.  =  the  ultimate  cohesive  strength  of  each  square  inch  of 
cross  sectional  area  in  the  section  PH,  supposing  the  centrifugal 
forces  —  F  and  +  F  to  cause  the  separation  in  this  section. 
(rt  —  r2)  I  =  area  of  parallelogram  GR. 

Hence,  the  force  in  Ibs.  required  to  cause  rupture  will  be, 

2  (r±  —  r3)  I  X  K ;  the  weight  of  the  stone  G  =  t  (r*  —  r*)  ?y, 
and  the  radius  of  gyration  of  each  half  of  the  stone,  i.  e.  the  distance 

4  y.  3  A.  3 

of  its  centre  of  gravity  from  the  axis  of  rotation  r  —  ^—  X  -2  __    32* 

At  the  moment  of  rupture,  the  centrifugal  force  of  half  the  stone 
is  equivalent  to  the  strength ;  we  hence  obtain  the  equation  of  con- 


452          THE  PRACTICAL  MODEL  CALCULATOR. 

dition  co  x  J  —  ^  =  2  (r4  -  rz)  IK,  i.  e.  «2  x  f  (/•/  -  ra3)  — 

9  g 

2  (rt  —  ra)  ZK;  or  leaving  out  2  ?  on  both  sides,  it  follows  that 
t  ~  r.)  K 


/%  (rt  ~ 

"         3  -  ?* 


r  '  +  r,  r,  +  r/)  / 

If  r±  =  2  feet  =  24  inches,  rs  =  4  inches,  K  =  750  Ibs.,  and 
the  specific  gravity  of  the  millstone  =  2-5  ;  therefore  the  weight 

62*5  x  2-5 
of  a  cubic  inch  of  its  mass  =  —  -      —  =  0-0903  Ibs.  ;  it  follows 


that  the  angular  velocity  at  the  moment  of  rupture  is, 

/3  x  12  x  32-2  x  750         /869400 
~  W        688  x  0-9903         "  W02-1264  = 

If  the  number  of  rotations  per  minute  =  n,  we  have  then  w  = 
2  it  n                                       30  co                     30  x  112-1 
gQ    ;  hence,  inversely,  n  =         ,  but  here  = =  1070. 

The  average  number  of  rotations  of  such  a  millstone  is  only  120, 
therefore  9  times  less. 

With  what  velocity  must  a  body  of  8  Ibs.  impinge  against  an- 
other at  rest  of  25  Ibs.,  in  order  that  the  last  may  have  a  velocity 
of  2  feet  ?  Were  the  bodies  inelastic,  we  should  then  have  to  put : 

M±c±  8  x  c±  33 

v  —  TUT — n\F>  i.  e.  2  =  Q  _,   Or,  hence  c.  —  —r  =  8J  feet,  the  re- 

Ml   T   M3 

quired  velocity ;  but  were  they  elastic,  we  should  have  va  —  */r~~L^r  ; 

33 
hence,  c±  =  -g-  =  4 J-  feet. 

If  in  a  machine,  16  blows  per  minute  take  place  between  two  in- 

1000  1200 

elastic  bodies  M4  = Ibs.  and  M3  = Ibs.,  with  the  velo- 
cities c.  =  5  feet,  and  c2  =  2  feet,  then  the  loss  in  mechanical  ef- 

16       (5  -  2)2       1000-1200 
feet  from  these  blows  will  be  :  L  =  gQ  X  — ^ —  x  — 2200 —  ~ 

4  1         6000  400 

TC  X  ^  X  ^7-7  X  -^pr~  =  0*576  x  -y^-=  20-94  units  of  work  per 

15  b4'4         11  II 

second. 

If  two  trains  upon  a  railroad  of  120000  Ibs.  and  160000  Ibs. 
weight,  come  into  collision  with  the  velocities  c±  =  20,  and  <?,  = 
15  feet,  there  will  ensue  a  loss  of  mechanical  effect  expended  upon 
the  destruction  of  the  locomotives  and  carriages,  which  in  the  case 
of  perfect  inelasticity  of  the  impinging  parts,  will  amount  to 

(20  +  15)2      120000  x  160000  _  1         19_20000  _ 

~~%~  280000  <  64-4  >     ~28 

1344000  ft.  Ibs.,  or  units  of  work. 


453 


SHIP-BTJILDING  AND  NAVAL  ARCHITECTURE, 

Two  rules,  by  which  the  principal  calculations  in  the  art  of  ship- 
building are  made,  may  be  employed  to  measure  the  areva  or  super- 
ficial space  enclosed  by  a  curve,  and  a  straight  line  taken  as  a  base. 

RULE  I. — If  the  area  bounded  by  the  curve  line  ABC  and  the 
straight  line  AC  is  required  to  be  estimated,  by  the  rule,  the  base 
AC  is  divided  into  an  even  number  of  equal  parts,  to  give  an  odd 
number  of  points  of  division. 


2  3  4  5  6  7   8   9  10  11  12  13  14  15  16  17  18  19  20  21 C 


Where  the  base  AC  is  divided  into  twenty  equal  parts,  giving 
twenty-one  points  of  division,  and  the  lines  1*1,  2-2,  3-3,  &c.,  are 
drawn  from  these  points  at  right  angles  or  square  to  AC,  to  meet 
the  curve  ABC,  these  lines,  1*1,  2-2,  3-3,  &c.,  are  denominated  or- 
dinates,  and  the  linear  measurement  of  them,  on  a  scale  of  parts, 
is  taken  and  used  in  the  following  general  expression  of  the  rule. 

Area  =  {A  +  4  P  +  2  Q}  g. 

Where  A  =  sum  of  the  first  and  last,  ordinates,  or  1*1  and  21*21. 
4  P  =  sum  of  the  even  ordinates  multiplied  by  4. 

Or,  {2d  +  4th  +  6th  +  8th  +  10th  +  12th  +  14th  -f  16th  -f 
18th  4-  20th}  x  4. 

2  Q  =  sum  of  the  remaining  ordinates  ;  or, 

{3d  +  5th  +  7th  -f  9th  +  llth  -f  13th  +  15th  +  17th  -f 
19th}  x  2. 

And  r  is  equal  to  the  linear  measurement  of  the  common  inter- 
val between  the  ordinates,  or  one'of  the  equal  divisions  of  the  base 
AC.  This  rule,  for  determining  the  area  contained  under  the  curve 
and  the  base,  may  be  put  under  another  form  ;  for  as  the 


Area 


4P-f-2Q}  x;  it  may  be  transferred  into 


Area 


The  practical  advantages  to  be  derived  from  this  modification  of 
the  general  rule  will  appear  when  the  methods  of  calculation  are 
further  developed. 


454  THE   PKACTICAL   MODEL   CALCULATOR. 

B 


Al 


10      11     12      13     14      15      16  C 


RULE  II. — If  the  base  AC  be  so  divided  that  the  equal  intervals 
are  in  number  a  multiple  of  the  numeral  3,  then  the  total  number 
of  the  points  of  division,  and  consequently  the  ordinates  to  the 
curve,  will  be  a  multiple  of  the  numeral  3  with  one  added,  and  the 
area  under  the  curve  ABC,  and  the  base  AC,  can  be  determined 
by  the  following  general  expression  : 

3r 
Area  =  {A  +  2  P  -f  3  Q}  X  -g-. 

Where  A  =  sum  of  the  first  and  last  ordinates,  or  1  and  16. 

2  P  =  sum  of  the  4th,  7th,  10th,  13th,  multiplied  by  2,  or  ordi- 
nates bearing  the  distinction  of  being  in  position  as  multiples  of 
the  numeral  3,  with  one  added. 

3  Q,  the  sum  of  the  remaining  ordinates,  multiplied  by  3,  or  of 
the  2d,  3d,  5th,  6th,  7th,  8th,  9th,  llth,  12th,  14th,  and  15th, 
multiplied  by  3. 

The  number  of  equal  divisions  for  this  rule  must  be  either  3,  6, 
9,  12,  or  15,  &c.,  being  multiples  of  the  numeral  3,  whence  the  or- 
dinates will  be  in  number  under  such  divisions,  multiples  of  the 
numeral  3,  with  one  added. 

This  rule  admits  also  of  a  modification  in  form,  to  make  it  more 
convenient  of  application. 

3 
For  area  =  {A  +  2  P  +  3  Q}  x  g  r. 

As  before  advanced  for  the  change  adopted  in  the  general  ex- 
pression for  the  first  rule,  the  utility  of  this  modification  of  the 
second  rule  will  be  observable  when  the  calculations  on  the  im- 
mersed body  are  proceeded  with. 

The  rules  are  formed  under  the  supposition  that  in  the  first  rule 
the  curve  ABC,  which  passes  through  the  extremities  of  the  ordi- 
nates, is  a  portion  of  a  common  parabola,  while  in  the  second  rule 
the  curve  is  assumed  to  be  a  cubic  parabola ;  the  results  to  be  ob- 
tained from  an  indiscriminate  use  of  either  of  these  rules,  diifer 
from  each  other  in  so  trifling  a  degree,  (considered  practically  and 
not  mathematically,)  as  not  to  sensibly  affect  the  deductions  derived 
by  them. 

William  O'Neill,  or,  as  English  writers  term  him,  William  Neal, 
was  the  first  to  rectify  a  curve  of  any  sort ;  this  curve  was  the 
semi-cubical  parabola ;  these  rules,  of  such  use  in  the  art  of  ship- 
building, were  first  given  by  him,  but  as  is  usual,  claimed  by  Eng- 
lish pretenders. 

The  foregoing  rules,  when  applied  to  the  measurement  of  the 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  455 

immersed  portion  of  a  floating  body,  as  the  displacement  of  a  ship, 
are  used  as  follows. 

The  ship  is  considered  as  being  divided  longitudinally  by  equi- 
distant athwartship  or  transverse  vertical  planes,  the  boundaries 
of  which  planes  give  the  external  form  of  the  vessel  at  the  respec- 
tive stations,  and  therefore  the  comparative  forms  of  any  inter- 
mediate portion  of  it. 


L 


If  the  ship  be  immersed  to  the  line  AB,  considered  as  the  line 
of  the  proposed  deepest  immersion  or  lading,  the  curves  HLO  and 
KMF  would  give  the  external  form  of  the  ship  at  the  positions  G 
and  I  in  that  line  ;  and  the  areas  GHLO,  IKMF  contained  under 
the  curves  HLO,  KMF,  the  right  lines  GH,  IK,  (the  half-breadths 
of  the  plane  of  proposed  flotation  AB  at  the  points  G  and  I,)  and 
the  right  lines  GO,  IF,  the  immersed  depths  of  the  body  at  those 
points  are  the  areas  to  be  measured ;  and  if  the  areas  obtained  be 
represented  by  linear  measurements,  and  are  set  off  on  lines  drawn 
at  right  angles  to  the  line  AB  at  their  respective  stations,  a  curve 
bounding  the  representative  areas  would  be  formed,  and  the  mea- 
surement by  the  rules  of  the  area  contained  under  this  curve,  and 
the  right  line,  AB,  or  length  of  the  ship  on  the  load-water  .line, 
would  give  the  sum  of  the  areas  thus  represented,  and  thence  the 
solid  contents  of  the  immersed  portion  of  the  ship  in  cubic  feet  of 
space.  In  accordance  with  this  application  of  those  rules  to  mea- 
sure the  displacement  of  the  ship,  the  usual  practice  is  to  divide 
the  ship  into  equidistant  vertical  and  longitudinal  planes,  the  lon- 
gitudinal planes  being  parallel  to  the  load-water  section  or  hori- 
zontal section  formed  by  the  proposed  deepest  immersion. 

To  measure  the  areas  of  these  planes  after  they  have  been  de- 
lineated by  the  draughtsman,  the  constructor  divides  the  depth  of 
each  of  the  vertical  sections,  or  the  length  of  each  horizontal  sec- 
tion, into  such  a  number  of  equal  divisions  as  will  make  either  one 
or  the  other  of  the  rules  1  or  2  applicable.  If  the  first  rule  be 
preferred,  the  equal  divisions  must  be  of  an  even  number,  so  that 
there  may  be  an  odd  number  of  ordinates ;  while  the  use  of  the 
second  rule,  to  measure  the  area,  will  require  the  equal  divisions 
of  the  base  to  be  in  number  a  multiple  of  the  numeral  3,  which 
will  make  the  ordinates  to  be  in  number  a  multiple  of  the  numeral 
3,  with  one  added.  From  the  points  of  equal  divisions  in  the  re- 
spective sections  thus  determined,  perpendicular  ordinates  are 
drawn  to  meet  the  curve,  or  the  external  form  of  the  transverse 
planes  of  the  body;  and  a  table  for  the  ordinates  thus  obtained, 
having  been  made,  as  shown  page  467,  the  measures  by  scale  of  the 
respective  ordinates  are  therein  inserted. 


456  THE    PRACTICAL    MODEL    CALCULATOR. 

For  the  area  IKMF,  the  linear  measurements  of  IK,  1-1,  2-2, 
3-3,  4-4,  are  taken  by  a  scale  of  parts,  and  inserted  in  the  column 
marked  5,  page  467,  the  whole  length  AB  of  the  load-water  line 
being  divided  into  10  equal  divisions,  and  the  area  IKMF  being 
supposed  as  the  fifth  from  B,  the  fore  extreme  of  the  load-water 
line.  To  apply  the  first  rule  to  the  measurement  of  the  area  of  No. 
5  section,  the  ordinates  are  extracted  from  the  table,  page  467,  and 
operated  upon  as  directed  by  the  rule  ;  viz. 

Area  =  {  A  +  4  P  -f  2  Q}  X  g. 

IK,  or  first,                     1-1,  or  2d,  2-2  or  3d, 

4-4,  or  last,                      3-3,  or  4th,  X  2. 

added  together  or  2  Q. 
added  together  =  A.        and  X  4  =  4  P. 

By  rule,  area  =  {A  +-  4  P  -f  2  Q}  x  g. 
Whence  area  =  {(IK  -f  44)  +  (1-1  +  3-3)  4  -f  2-2  x  2}  x  g  = 

area  IKMF  ;  and,  in  a  similar  manner,  may  the  several  areas  of 
the  other  transverse  sections  be  determined. 

When  these  areas  have  all  been  thus  measured,  they  are  to  be 
summed  by  the  same  rules  ;  the  areas  themselves  being  considered 
as  lines,  and  the  result  will  give  the  solid  for  displacement  in  cubic 
feet.  To  shorten  this  tedious  application  of  the  formula,  the  ar- 
rangement of  having  double-columned  tables  of  ordinates  was  in- 
troduced, as  shown  on  page  484,  and  for  the  more  ready  use  of  this 
enlarged  table,  the  modifications  in  the  formula  467,  before  alluded 
to,  were  adopted,  that  of 


Area  =      A  +  4P  +  2QXg  =      2--f2P-fQ       X  -, 
and  that  of 

Area=      A  +  2P-f3Qx:  =          +  P  +  1-5  Q  1  x|rT 


as  rendering  the  required  number  of  figures  much  less,  whereby 
accuracy  of  calculation  is  insured  and  time  is  saved. 

In  using  a  table  of  ordinates  constructed  for  this  method  of  cal- 
culation, the  linear  measurement  of  the  several  ordinates  of  vertical 
section  5  and  the  corresponding  ones  of  all  the  others  would  be  in- 
serted in  the  double  columns  prepared  for  them,  in  the  following 
order  :  — 

In  the  first  and  last  lines  of  the  enlarged  table  for  the  ordinates, 

A 

distinguishable  by  -^r,  in  the  left-hand  column  of  each  pair,  the 

measurements  of  the  first  and  last  ordinates  of  the  respective  areas 
are  placed,  and  in  the  right-hand  column  of  each  pair  one-half  of 
such  measurements,  as  being  one-half  of  the  first  and  last  ordinates 
of  each  vertical  section  or  area.  In  the  lines  distinguished  by  2  P, 
in  the  left-hand  column,  the  measurements  of  the  even  ordinates 


SHIP-BUILDING   AND   NAVAL   ARCHITECTURE.  457 

of  each  respective  area  are  placed,  which  having  been  multiplied 
by  two,  the  result  is  placed  in  the  respective  right-hand  columns 
prepared  for  each  vertical  section  ;  while  in  those  lines  of  the  table 
distinguished  by  Q,  the  measurements  of  the  ordinates  themselves 
are  placed  in  the  right-hand  columns,  as  not  requiring  by  the  modifi- 
cation of  the  rules  any  operation  to  be  used  on  them,  before  being 
taken  into  the  sum  forming  the  sub-multiple  of  the  respective  areas. 

It  may  here  with  propriety  be  suggested,  that  in  practice  the 
insertion  of  the  linear  measurements  of  the  ordinates  in  the  table 
in  red  ink  will  be  found  useful,  and  that  after  such  has  been  done, 
by  the  upper  line  of  figures  in  the  table  of  ordinates  thus  arranged, 
being  divided  by  two,  the  second  line  of  figures  being  multipled 
by  two,  and  so  on  with  the  others  as  shown  by  the  table,  and  the 
results  thus  obtained  being  inserted  in  their  respective  right-hand 
columns  as  before  described,  great  facility  and  despatch  of  calcu- 
lation are  afforded  to  the  constructor. 

That  this  method  will  yield  a  correct  measurement  of  the  areas 
will  be  evident  by  an  inspection  of  the  terms  of  the  general  expres- 
sion of  area  =  <-o4-2P+Q>  X  -5-,  which  are  placed  against 

the  several  lines  of  the  table  of  ordinates.  And  it  will  be  equally 
apparent,  that  the  sum  total  of  the  figures  inserted  in  the  right- 
hand  columns  appropriated  to  each  section  is  a  sub-multiple  of 
the  area  of  each  section,  and  that  these  results  arising  from  the 

{A  1 

•p  +  2  P  +  Q  >  will  be  one-half  of 

those  that  would  be  obtained  by  abstracting  the  ordinates  from  the 
table,  page  467,  and  using  them  in  the  expression  A  -f  4  P  -f  2  Q  ; 
and  therefore  to  complete  the  calculation  for  the  areas  by  the  rule, 

2r 
the  first  results  for  the  areas  must  be  multiplied  by  -«-,  and  the 

last  by  oj  where  r  is  equal  to  the  common  interval  or  equal  divi- 
sion of  the  base  in  linear  feet ;  or  the  part  of  the  expression  for 
{A  i  2  T 

~2  +  2  P  -f  Q  >  must  be  multiplied  by  -g-,  to  make  it 

equivalent  to  {A  -f  4P  -f  2  Q}  X  g. 

The  sub-multiples  of  the  areas  of  the  vertical  sections  thus  deter- 
mined, require  to  be  summed  together  for  the  solid  of  displacement, 
and  by  considering  the  sub-multiples  of  the  areas  to  be,  as  before 
stated,  represented  by  lines  or  proportionate  ordinates,  O'Neill's 
rules,  by  the  same  table  of  ordinates  with  an  additional  column,  may 
be  made  available  to  the  development  of  the  solid  of  displacement. 
For  the  sectional  areas  being  represented  by  lines,  by  the  first  rule, 
one-half  the  first  and  last  areas,  added  to  the  sum  of  the  products 
arising  from  multiplying  the  even  ordinates  or  representative  areas 
by  two,  together  with  the  odd  ordinates  or  the  areas  as  given  by 


458  THE    PRACTICAL    MODEL   CALCULATOR. 

the  tables,  and  these  being  placed  in  the  additional  column  of  the 
table  prepared  for  them,  the  sub-multiple  of  the  solid  of  displace- 
ment will  be  given. 

The   operation  will  stand  thus:    Sub-multiple   of  each  of  the 

areas  =  ^-^-fSP  +  QLor  each  area  will  be  -g-  less  than  the 

full  result,  and  the  representative  lines  for  the  areas  will  be  dimi- 
nished in  that  proportion  ;  and  having  used  these  sub-multiples  of 
the  areas  thus  diminished  in  the  second  operation  for  obtaining  the 
sub-multiple  of  the  solid  of  displacement  under  the  same  rule,  the 

2r' 
results  will  again  be  -5-  less  than  the  true  result  ;  therefore  the 

sum  thus  determined  will  have  to  be  multiplied  by  the  quantity 


IT  X  —  q-,   to  give  the  solid  required.      In  this   expression,   of 

2r      2r' 

~o~  X  —  o-,  T  =  the  equal  distances  taken  in  the  vertical  planes  to 

obtain  the  respective  vertical  areas  ;  rf  =  the  equal  distances  at  which 
the  vertical  areas  are  apart  on  the  longitudinal  plane  of  the  ship. 

The  displacement  being  thus  determined,  by  an  arrangement  of 
an  enlarged  table  of  ordinates,  the  functions  arising  from  the  sub- 
multiples  of  the  areas  of  the  vertical  sections  being  placed  in  O'Neill's 
rules  to  ascertain  the  displacement,  may  be  used  in  the  table  of 
ordinates  to  find  the  distance  of  the  centre  of  gravity  of  the  im- 
mersed body  from  any  assumed  vertical  plane  ;  and  also  the  dis- 
tance that  the  same  point  —  "  the  centre  of  gravity  of  displacement" 
—  is  in  depth  from  the  load-water  or  line  of  deepest  immersion,  and 
that  from  the  considerations  which  follow  :  — 

In  a  system  of  bodies,  the  centre  of  gravity  of  it  is  found  by 
multiplying  the  magnitude  or  density  of  each  body  by  its  respective 
distance  from  the  beginning  of  the  system,  and  dividing  the  sum 
of  such  products  by  the  sum  of  the  magnitudes  or  densities.  The 
displacement  of  a  ship  may  be  considered  as  made  up  of  a  suc- 
cession of  vertical  immersed  areas  ;  and  if  it  be  assumed  that  the 
moments  arising  from  multiplying  the  area  of  each  section  by  its 
relative  distance  from  an  initial  plane  may  be  represented  by  suc- 
cessive lineal  measurements,  the  general  rules  will  furnish  the  sum- 
mation of  such  moments  ;  and  the  displacement  or  sum  of  the  areas 
has  been  obtained  by  a  similar  process,  from  whence,  by  the  rule 
for  finding  the  centre  of  gravity  of  a  system  as  before  given,  the 
distance  of  the  common  centre  of  gravity  from  the  assumed  initial 
plane  would  be  ascertained,  by  dividing  the  sum  of  the  moments  of 
the  areas  by  the  sum  of  the  areas  or  the  solid  of  displacement. 

To  extend  this  reasoning  to  the  enlarged  table  of  ordinates  used 
for  the  second  method  of  calculation  :  The  sub-multiples  of  the 
respective  areas,  when  put  into  the  formulas  to  obtain  the  propor- 
tionate solid  of  displacement,  are  relatively  changed  in  value  to 
give  that  solid,  and  consequently  the  moments  of  such  functions  of 


SHIP-BUILDING   AND   NAVAL  ARCHITECTURE.  459 

the  vertical  areas  will  be  to  each  other  in  the  same  ratio ;  and  the 
sum  of  these  proportionate  moments,  if  considered  as  lines,  can  be 
ascertained  by  multiplying  the  functions  of  the  areas  by  their  rela- 
tive distances  from  the  assumed  initial  plane,  or  by  the  number  of 
the  equal  intervals  of  division  they  are  respectively  from  it,  and 
afterwards,  by  the  rules,  summing  these  results,  forming  the  sum 
of  the  moments  of  the  sub-multiples  of  the  functions  of  the  verti- 
cal areas :  and  the  proportionate  sub-multiple  for  the  displacement 
is  shown  on  the  table ;  the  division  therefore  of  the  former,  or  the 
sum  of  the  proportional  moments  of  the  functions  of  the  areas,  by 
the  proportionate  sub-multiple  for  the  displacement,  will  give  the 
distance  (in  intervals  of  equal  division)  that  the  centre  of  gravity 
of  the  displacement  is  from  the  initial  plane,  which  being  multi- 
plied by  the  value  in  feet  of  the  equal  intervals  between  the  areas, 
will  give  the  distance  in  feet  from  the  assumed  initial  plane,  or  from 
the  extremity  of  the  base  line  of  the  proportional  sectional  areas 
for  displacement.  This  reasoning  will  apply  equally  to  finding  the 
position  of  the  centre  of  gravity  of  the  body  immersed,  both  as 
respects  length  and  depth,  and  on  the  enlarged  tables  for  construc- 
tion given,  (pages  484  and  485,)  the  constructor,  by  adopting  this 
arrangement,  will  at  once  have  under  his  observation  the  calcula- 
tions on,  and  the  results  of,  the  most  important  elements  of  a  naval 
construction. 

The  foregoing  tabular  system,  for  the  application  of  O'Neill's 
rules  to  the  calculations  required  on  the  immersed  volume  of  a 
ship's  bottom,  led  to  a  lineal  delineation  of  the  numerical  results 
of  the  tables,  and  thence  the  development  of  a  curve  of  sectional 
areas,  on  a  base  equivalent  to  the  length  of  the  immersed  portion 
of  the  body,  or  of  the  length  at  the  load-water  line.  To  effect 
this,  the  sub-multiples  of  the  sectional  areas,  taken  from  the  tables 
for  calculation,  are  severally -divided  by  such  a  constant  number  as 
to  make  their  delineation  convenient;  then  these  thus  further 
reduced  sub-multiples  of  the  areas,  being  set  off  at  their  respective 
positions  on  the  base,  formed  by  the  length  of  the  load-water  line, 
a  curve  passed  through  the  extreme  points  of  these  measurements, 
will  bound  an  area,  that  to  the  depth  used  for  the  common  divisor 
would  form  a  zone,  representative  of  the  solid  of  displacement? 
The  accuracy  of  such  a  representation  will  be  easily  admitted,  if 
the  former  reasoning  is  referred  to. 

To  obtain  the  solid  of  displacement  from  this  representative  area, 
the  load-water  line  or  plane  of  deepest  immersion  is  considered  as 
being  divided  lengthwise  into  two  equal  parts,  which  assumption 
divides  the  base  of  the  curve  of  sectional  areas  also  into  two  equal 
portions :  the  line  of  representative  area  to  that  medial  point  is 
then  drawn  to  the  curve,  and  triangles  are  formed  on  each  side  of 
it  by  joining  the  point  where  it  meets  the  curve  with  the  extremi- 
ties of  the  base  line ;  this  arrangement  divides  the  representative 
area  into  four  parts,  two  triangles  which  are  equal,  viz.  1  and  2, 
and  two  other  areas  which  are  contained  under  the  hypothenuse  of 


460          THE  PRACTICAL  MODEL  CALCULATOR. 

these  triangles  and  the  curves  of  sections,  or  3  and  4  of  the  an- 
nexed diagram. 

Diagram  of  a  Curve  of  Sectional  Areas. 


A  D  c 

ABCDA  equal  sectional  area,  representative  of  the  half  displace- 
ment as  a  zone  of  a  given  common  depth. 

AC  equal  the  length  of  the  load-water  section  from  the  fore-side 
of  the  rabbet  of  the  stem  to  the  aft-side  of  the  rabbet  of  the  post, 
and  D  the  point  of  equal  division. 

BD,  the  representative  area  of  half  the  immersed  vertical  sec- 
tion at  the  medial  point  D,  joining  B  with  the  points  A  and  C,  will 
complete  the  division  of  the  representative  area  ABCDA. 

ABD  and  CBD,  under  such  considerations,  are  equal  triangles. 

BECB,  BFAB,  areas,  bounded  respectively  by  the  hypothenuse 
AB  or  BC  of  the  triangles  and  the  curve  of  sectional  areas  ;  and, 
supposing  the  curves  AFB  and  BEC  to  be  portions  of  common 
parabolas,  the  solid  of  displacement  will  be  in  the  following  terms  : 

The  area  of  each  of  the  triangles  is  equal  to  J  of  AC  X  BD  ; 
hence  the  sum  of  the  two  =  J  of  AC  x  BD  :  the  hypothenuse  AB 

or  BC  =  J[(~2")    +  BD2]>  and  the  area  of  BECB  if  consi- 

dered as  approximating  to  a  common  parabola  =  J  [_  \2~)   ~^~  "^  J 
X  f  of  the  greatest  perpendicular  on  the  hypothenuse  BC. 
Area  of  BFAB  under  the  same  assumption  =  J  [  \~~9~  )   ~^~  BD2  J 

x  f  of  the  greatest  perpendicular  on  the  hypothenuse  AB  ;  whence 
the  whole  displacement  will  be   expressed  by  J  AC    X   BD    X 

jfv~«~)    +  BD2J  x  f  of  the  greatest  perpendicular  on  the  hypo- 
thenuse B  C  +        (--      +  BD2     x  -f  of  the  greatest  perpendi- 


cular on  the  hypothenuse  AB. 

By  a  similar  method,  from  the  light  draught  of  water,  or  the 
depth  of  immersion  on  launching  the  ship,  the  light  displacement, 
or  the  weight  of  the  hull  or  fabric,  may  be  delineated  and  esti- 
mated ;  and  the  representative  curve  for  it  being  placed  relatively  on 
the  same  base  as  that  used  for  the  representative  curve  for  the  load 
displacement,  the  area  contained  between  the  curve  bounding  the 
representative  area  for  the  load  displacement,  and  the  curve  bound- 
ing the  representative  area  for  the  light  displacement,  will  be  a  repre- 
sentative area  of  the  sum  of  the  weights  to  be  received  on  board, 
and  point  out  their  position  to  bring  the  ship  from  the  light  line 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE. 


461 


of  flotation,  or  the  line  of  immersion  due  to  the  weight  of  the  hull 
when  completed  in  every  respect,  to  that  of  the  deepest  immersion, 
or  the  proposed  load-water  line  of  the  constructor — a  representa- 
tion that  would  enable  the  constructor  to  apportion  the  weights  to 
be  placed  on  board  to  the  upward  pressure  of  the  water,  and  thence 
approximate  to  the  stowage  that  would  insure  the  easiest  movements 
of  a  ship  in  a  sea. 

By  an  inspection  of  the  diagram  of  the  curve  of  sectional  areas, 
it  will  clearly  be  seen  that  the  representative  area  for  displacement 
under  the  division  of  it,  into  the  triangles  1  and  2,  and  parabolic 
portions  of  the  area  3  and  4,  will  point  out  the  relative  capacities 
of  the  displacement,  under  the  fore  and  after  half-lengths  of  the 
base  or  load-water  line ;  for,  by  construction,  the  triangles  ABD 
and  CBD  are  equal,  and  therefore  the  comparative  values  of  the 

areas  BECB  and  BFAB,  or  of  ^[(-g-)  +  BD2]  x  f  of  the 
greatest  perpendicular  on  the  hypothenuse  BC,  compared  with 
J  [>  (^~2~  )  +  BD2  J  X  f  of  the  greatest  perpendicular  on  the  hypo- 
thenuse AB,  or  of  the  relative  values  of  the  greatest  perpendicu- 
lars on  the  hypothenuses  BC  and  AB,  will  give  the  relative  capaci- 
ties of  the  fore  and  after  portions  of  the  immersed  body  or  the  dis- 
placement. 

The  representative  area  ABCDA  admits  also  of  a  measurement 
by  the  second  rule. 

Let  BD,  as  before,  be  the  representative  area  at  the  middle  point. 


A6 


Divide  AD  or  DC  into  three  equal  portions,  then  the  equal  divi- 
sions being  a  multiple  of  3,  the  second  rule  is  applicable  to 
measure  the  areas  ABDA  or  BCDB  ;  for  the  area  ABDA  = 


6,6  +  BD  -f-  2  x  0  +  3  {4,4  -f  5,5}      ~-  6,6  =  0  ; 
|  BD  -f  3  {4,4  +  5,5}  j-g-  ;  and  area  BCDB  = 


1,1  +  BD  +  2  x'O  +  3  x  {2,2  -f  3,3}     -,  where  1,1=0 
/  BD  +  3  X  {2,2  -f  3,3}  j-g-  =  BCDB,  and  the  displace- 
ment =  |  BD  -f  3  x  {4,4  -f  5,5}  1  -g-  +  j  BD  -f  3  x  {2,2  -f  3,3}  1 


AD 


X  -g-  x  by  the  constant  divisor  of  the  areas,  or  the  depth  of  the 
zone  in  feet. 


462          THE  PEACTICAL  MODEL  CALCULATOR. 

The  rules  given  by  O'Neill  for  the  measurement  of  the  im- 
mersed portion  of  the  body  of  a  ship,  having  been  theoretically 
stated,  the  practical  application  of  them  will  be  given  on  the  con- 
struction. 

The  immersed  part  of  a  ship,  being  a  portion  of  the  parallelopi- 
pedon  formed  by  the  three  dimensions ; — length  on  the  load-water 
line,  from  the  foreside  of  the  rabbet  of  the  stem  to  the  aftside 
of  the  rabbet  of  the  stern-post ;  extreme  breadth  in  midships  of  the 
load-water  section ;  and  depth  of  immersion  in  midships  from  the 
lower  edge  of  the  rabbet  of  the  keel ; — it  would  seem  that  the  first 
step  towards  the  reduction  of  the  parallelopipedon,  or  oblong,  into 
the  required  form,  would  be  to  find  what  portion  of  it  would  be  of 
the  same  contents  as  the  proposed  displacement  of  the  ship — a 
knowledge  of  which  would  enable  the  constructor,  by  a  comparison 
of  the  result  with  a  similar  element  of  an  approved  ship,  to  deter- 
mine whether  the  principal  dimensions  assumed  would  (under  the 
form  intended)  give  an  immersed  body  equal  to  carrying  the  pro- 
posed weights  or  lading. 

The  relative  capacities  of  the  immersed  bodies  contained  under 
the  fore  and  after  lengths  of  the  load-water  line  must  next  be  fixed, 
and  the  constructor  in  this  very  important  element  of  a  construc- 
tion will  find  little  to  guide  him  from  the  results  of  past  experience 
and  practice.  From  deductions  on  approved  ships  of  rival  con- 
structors it  will  be  developed,  that  in  this  essential  element,  "  the 
relative  difference  between  the  two  bodies,"  they  vary  from  1  to 
13  per  cent,  on  the  whole  displacement. 

The  relative  capacities  of  the  fore  and  after  bodies  of  immersion 
under  the  proposed  load-water  line  would  seem  at  the  first  glance 
of  the  subject  to  be  a  fixed  and  determinate  quantity,  as  being  a 
conclusion  easily  arrived  at  from  a  knowledge  of  the  proportions 
due  to  the  superincumbent  weights — under  such  a  consideration,  the 
weight  of  the  anchors,  bowsprit,  and  foremast  would  necessarily  be 
supposed  to  require  an  excess  in  the  body  immersed  under  the  fore 
half-length  of  the  load-water  line  over  that  immersed  under  the 
after  half-length  of  the  same  element. 

In  a  ship,  the  necessary  arrangement  of  the  weights,  to  preserve 
the  proposed  relative  immersion  of  the  extremes  or  the  intended 
draught  of  water,  would  be  pointed  out  by  a  delineated  curve  of 
sectional  areas,  described  as  before  directed ;  but  a  want  of  that 
system,  or  of  some  other,  has  often  caused  an  error  in  the  actual 
draught  of  water,  and  that  under  a  great  relative  excess  of  the 
volumes  of  displacement  in  the  fore  and  after  portions  of  the  im- 
mersed body. 

The  men-of-war  brigs  built  to  a  construction-draught  of  water 
12  ft.  9  in.  forward,  14  ft.  4  in.  abaft,  giving  1  ft.  7  in.  difference, 
had  under  such  a  construction  a  difference  of  displacement  between 
the  immersed  bodies  under  the  fore  and  after  half-lengths  of  the 
load-water  line  that  was  equivalent  to  10*4  tons  for  every  100  tons 
of  the  vessel's  total  displacement  or  weight;  but  these  ships,  when 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  463 

stowed  and  equipped  for  sea,  came  to  the  load-draught  of  water  of 
14  ft.  2  in.  forward,  14  ft.  3  in.  aft, — difference  1  inch,  or  an  immer- 
sion of  the  fore  extreme  of  18  inches  more  than  was  intended  by 
the  constructor.  The  reason  of  this  practical  departure  from  the 
proposed  line  of  flotation  of  the  constructor  was,  that  the  inter- 
nal space  or  hold  of  the  ship  necessarily  followed  the  external  form, 
giving  a  hold  proportionate  to  the  displacement  contained  under  the 
several  portions  of  the  body ;  but  an  injudicious  disposal  of  the 
stores  (in  placing  the  weights  too  far  forward)  made  them  more  than 
equivalent  to  the  upward  pressure  of  the  water  at  the  respective 
portions  of  the  proposed  immersion  of  the  body,  and  thence  arose 
the  error  or  excess  in  the  fore  immersion  by  giving  a  greater  draught 
of  water  than  was  designed.  The  stowage  of  a  ship's  hold,  under 
a  reference  to  the  representative  area  for  the  displacement,  con- 
tained between  the  curves  of  sectional  areas  developed  for  the  light 
and  load  displacements,  would  prevent  similar  errors  under  any 
extent  to  which  the  relative  capacity  of  the  two  bodies  might  be 
carried.  This  relative  capacity  of  the  two  bodies  will  affect  the 
form  of  the  vessel's  extremes,  giving  a  short  or  long  bow,  a  clear 
or  full  run  to  the  rudder ;  for  the  whole  displacement  being  a  fixed 
quantity,  if  the  portion  of  it  under  the  fore  half-length  of  the  load- 
water  line  be  increased,  it  must  be  followed  by  a  proportionate  dimi- 
nution of  the  portion  of  the  displacement  under  the  after  half-length 
of  the  load-water  line,  so  that  the  total  volume  of  the  displacement 
may  remain  the  same,  which  arrangement  will  give  a  proportionately 
full  bow  and  clean  run,  and  vice  versd. 

The  curve  of  sectional  areas  under  the  foregoing  considerations 
is  also  applicable  to  a  comparison  of  the  relative  qualities  of  ships 
of  the  same  rate,  by  showing  at  one  view  the  distribution  of  the 
volume  of  displacement  in  each  ship,  under  the  draught  of  water 
which  has  been  found  on  trial  to  give  the  greatest  velocity ;  based 
on  which,  deductions  may  be  made  from  the  relative  capacities  of 
the  bodies  pointed  out  by  the  sectional  curves,  that  will  serve  to 
guide  the  naval  constructor  in  future  constructions. 

The  curve  of  sectional  areas  is  also  available  for  forming  a  scale 
to  measure  the  amount  of  displacement  of  a  ship  to  any  assumed 
or  given  draught  of  water.  To  effect  this,  on  the  sheer  draught  or 
longitudinal  plan  of  the  ship  between  the  load-water  line,  or  that 
of  deepest  immersion,  and  the  line  denoting  the  upper  edge  of  the 
rabbet  of  the  keel,  draw  intermediate  lines  parallel  to  the  load- 
water  line  as  denoting  lines  of  intermediate  immersion  between  the 
keel  and  load-water  line ;  these  lines  may  be  placed  equidistant 
from  each  other,  but  they  are  not  necessarily  required  to  be  so. 
Find  the  curve  of  sectional  areas,  due  to  each  immersion  of  the 
ship  denoted  by  these  lines,  and  measure  the  areas  bounded  respec- 
tively by  these  curves,  in  the  manner  as  before  directed  for  the  load 
displacement :  these  results  will  give  the  magnitudes  of  the  im- 
mersed portions  of  the  body  in  cubic  feet,  which  being  divided  by  35, 
the  mean  of  the  number  of  cubic  feet  of  salt  or  fresh  water  that 


464  THE   PRACTICAL   MODEL   CALCULATOR. 

are  equivalent  to  a  ton  in  weight,  will  give  their  respective  weights 
in  tons. 

Assume  a  line  of  scale  for  depth,  or  mean  draught  of  water,  the 
lower  part  of  which  is  to  be  considered  the  underside  of  the  false 
keel  of  the  ship,  and  set  off  on  this  line,  by  means  of  a  scale  of 
parts,  the  depths  of  the  immersions  at  the  middle  section  of  the 
longitudinal  plan  ;  draw  lines  (at  the  points  thus  obtained)  perpen- 
dicular to  this  assumed  line  for  depth  or  draught  of  water,  and 
having  determined  a  scale  to  denote  the  tons,  set  off  on  each  line 
by  this  scale  the  tons  ascertained  by  the  curves  of  sectional  areas 
to  be  due  to  the  respective  immersions  of  the  body;  then  a  curve 
passed  through  these  points  will  be  one  on  which  the  weights  in  tons 
due  to  the  intermediate  immersions  of  the  body  may  be  ascertained  ; 
or,  the  displacement  of  a  ship  to  the  mean  of  a  given  draught  may 
be  found  by  setting  up  the  mean  depth  on  the  scale,  showing  the 
draught  of  water  —  transferring  that  depth  to  the  curve  for  tonnage, 
and  then  carrying  the  point  thus  obtained  on  the  curve  for  tonnage 
to  the  scale  of  tons,  which  will  give  the  number  of  tons  of  displa  ce- 
ment to  that  depth  of  immersion  or  draught  of  water. 

Description  of  the  several  plans  to  be  delineated  by  the  draughts- 
man, previous  to  the  commencement  of  the  calculations. 

Sheer  Plan.  —  A  projection  of  the  form  of  the  vessel  on  a  longi- 
tudinal and  vertical  plane,  assumed  to  pass  through  the  middle  of 
the  ship,  and  on  which  the  position  of  any  point  in  her  may  be 
fixed  with  respect  to  height  and  length. 

Half  -breadth  Plan.  —  The  form  of  the  vessel  projected  on  to  a 
longitudinal  and  horizontal  plane,  assumed  to  pass  through  the  ex- 
treme length  of  ship,  and  on  which  the  position  of  any  point  in 
the  ship  may  be  fixed  for  length  and  breadth. 

Body  Plan.  —  The  forms  of  the  vertical  and  athwartship  sections 
of  the  ship,  projected  on  to   a  vertical  and  athwartship  plane, 
assumed  to  pass  through  the  largest  athwartship  and  vertical  sec- 
tion of  her,  and  on  which  plan  the  position  of  any  point  in  the  ship 
may  be  fixed  for  height  and  breadth. 

These  plans  conjointly  will  determine  every  possible  point  re- 
quired ;  for,  by  inspection,  it  will  be  found  — 
That  the  sheer  and  half-breadth  plans  have 

one  dimension  common  to  both,  viz.:  ..........  Length. 

Half-breadth  and  body  plane  .......................  Breadth. 

Sheer  and  body  plane  ................................  Height. 

For  sheer  plan  gives  length  and  height  ......  )    f  ., 

Half-breadth  plan  gives  length  and  breadth  V01 


Body  plan  gives  breadth  and  height  .........  J 

Which  dimensions  form  the  co-ordinates  for  any  point  in  the  solid, 
and  must  determine  the  position  of  it. 

The  point  C  in  the  load-water  section  AB,  has  for  its  co-ordi- 
nates to  fix  its  position, 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  465 

The  length,   1-5  of  the  half-breadth  plan. 

Height,  5*0  of  the  sheer  plan, 
And  the  breadth,  1*0  of  the  body  plan  of  section. 
And  the  same  for  any  other  point  of  the  solid  or  of  the  ship. 

In  the  sheer  plan,  AB  represents  the  line  of  deepest  immersion, 
a  a,  bb,  cc,  dd,  lines  drawn  parallel  to  that  line  at  a  distance  of 
•9  feet,  making  with  AB  an  odd  number  of  ordinates  for  the  use  of 

T 

the  first  general  rule  for  the  area,  where  area  =  {A  +  4P-f2Q}  x  o, 
and  A  =  the  sum  of  the  first  and  last  ordinates. 
P  =  the  sum  of  the  even  ordinates,  as  2,  4. 
Q  =  the  sum  of  the  odd  ordinates,  as  3,  &c. 

The  line  AB,  or  length  of  the  load-water  line,  is  bisected  at  C, 
and  AC,  CB  are  thence  equal;  C  being  the  middle  point  of  the 
load-water  line,  the  spaces  BC,  AC  are  again  divided  into  four 
equal  divisions,  giving  five  ordinates  for  each  space,  at  a  distance 
apart  of  5-5  feet. 

This  arrangement  will  give  the  immersed  body  of  the  vessel,  as 
being  divided  into  two  parts  under  an  equal  division  of  the  load- 
water  line,  and  an  odd  number  of  ordinates  in  each  section  of  the 
body  for  the  application  of  the  first  general  rule  given  for  finding 
the  areas  of  the  vertical  sections  and  thence  the  displacement. 

The  half-breadth  plan  delineates  the  form  of  the  body  immersed 
for  length  and  breadth,  the  line  AB  of  the  sheer  plan  being  repre- 
sented in  the  half-breadth  plan  by  the  line  marked  AB,  and 
a  a,  b  5,  c  c,  d  d,  of  the  sheer  plan  by  the  lines  similarly  distin- 
guished in  the  half-breadth  plan. 

The  body  plan  gives  the  form  of  the  body  in  the  depth,  the  lines 
distinguished  5-5  in  the  sheer  and  half-breadth  plans  being  in  the 
body  plan  developed  by  the  curve  5-5-5,  giving  the  external  form 
of  the  ship  at  the  section  5-5 ;  the  same  reasoning  applies  to  the 
other  divisions  of  the  load-water  line  AB. 


A  pile  of  400  Ibs.  weight  is  driven  by  the  last  round  of  20  blows 
of  a  500  Ibs.  heavy  ram,  falling  from  a  height  of  5  feet;  6  inches 
deeper,  what  resistance  will  the  ground  offer,  or  what  load  will  the 
pile  sustain  without  penetrating  deeper?  ~  ~ 

Here  G  =  400,  Gx  =  700  Ibs.,  H  =  5,  and  0  =  -^  =  0-025  feet, 

whereby  it  is   supposed  that  the  pile  penetrates  equally  far  for 
each  blow. 

/   700   \2400  X  5   /  7  x2 

~  0-025-  -  (n)  x  80000  =  3240°  !bs-> 


the  ram,  not  during  penetration,  remaining  upon  the  pile. 

7002  x  5          4900 

^  =  1100  x  0-025  =  ~TT~  X  ^  =  89100  ^s.,  the  ram  remain- 
ing upon  the  pile  during  penetration. 

For  duration,  with  security,  such  piles  are  only  loaded  from  r£5 
to  jig  of  their  strength. 

30 


466 


THE  PRACTICAL  MODEL  CALCULATOR. 


Principal  Dimensions. 

Ft.  In. 

Length  for  Tonnage 45  0 

Keel  for  Tonnage 36  lOf 

Breadth  for  do 13  6 

Bunhen  in  Tuns 35  If 


AB,  Load-water  Line. 

aa\ 

b  b  I  Lines  parallel  to  AB  at  the 

c  c   \     distance  of  "92  feet  apart. 

dd) 

Draught  of  Water. 

Ft,  In 

Afore 4   <> 

Abaft 7   0 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE. 


467 


Calculations  required  for  the  construction  draw- 
ing of  a  yacht  of  36  tons. — 1st.  Usual  mode  of 
calculating  the  displacement  by  vertical  and 
horizontal  sections. 

TABLE  of  Ordinates  for  Yacht  of  36  Tons. 


Distinguish-") 
ing  No.  of    \- 

1 

2 

3 

4 

(5) 

6 

7 

8 

9 

the  sections.  J 

1'  A 

4 

iT6 

5^0 

6-0 

6^3 

~^l 

54 

3^7 

~4 

r  —  the  distance  be- 
tween the  ordi- 

2'P 

."35 

24 

4^2 

O  *D 

5r6 

5^5 

44 

2^6 

^35 

nates  used  for 
the  vertical  sec- 









,_ 



,  





tion  =  -92  feet. 

3'Q 

•3 

1-7 

3-2 

445-0 

4-6 

34 

1-7 

•3 

r'  —  the  distance  be- 

tween the  ordi- 

4'P 

^25 

1-0 

2-2 

3^2 

3-8 

34 

24 

1-1 

^25 

nates  used  for 
the   horizontal 

' 

..  





—  '••' 



•'•'     • 

sections  —  5'5 

5'  A 

•2 

4 

1-3 

2-0 

2-4 

2^0 

14 

•6 

•2 

feet. 

From  this  Table  the  following  application  of 
O'Neill's  rule,  No.  1,  is  usually  made  to  obtain 
the  volume  of  displacement  to  the  draught  of 
water  shown  on  the  drawing  as  the  load-water 
line,  or  line  of  proposed  deepest  immersion,  de- 
signated by  AB. 

General  terms  of  the  rule  : — 


Area  =  -^A  +  4P  +  2 


To  find  J  the  area  of  vertical  section  1,  fore 
body : — 


A=sum  of  "|  *4 
the  first  > 
arid  last  j  -2 


4P=fourtimesthesum^  -35 
of  the  even  prdinates,  > 
or  of  (2)  and  (4) J  -25 


•6  =  A  -60=P 

4- 

2^=4? 

2  Q  =  twice  the  sum  of  the  odd  1         -3  =  Q 
ordinates,  or  of  (3)  /   X    2 

1)0  =  2Q 
Whence  the  area,  which  is  equal  to 


•92 
3-6  x  -g- 

section  1. 


1-2  x  -92  =  1-104  =  J  area  of 


Which  sum  is  half  the  area  of  the  section  1,  and  is  kept  in  that 
form    of   the    half-measurement   for  the  convenience   of   calcula- 

tion. 


468  THE   PRACTICAL   MODEL   CALCULATOR. 

FORE  BODY. 

Vertical  Section  2. 


3-0 

=  A 

2-4                                            1-7 
1-0                                               2 

34  =  P                                    34  =  2Q 
4 

13-6  =  4  P 
34  -A 
34  =  2  Q 

20-4  =  A  +  4P  +  2Q 
•92  =  r 

IBS 

1836 

3)18-768 

5-0 
1-3 
6-3 

6-0 
2-0 

8^ 

4-2 
2-2 
=  A       6-4 
4 

6-256  =  |  area  of  Section  2. 

Vertical  Section  3. 
3-2 
2 
=  P                                            64  =  2Q 

=  4P 
=  A 
=  2Q 
=  A  +  4P  +  2Q 

T 

25-6 
6-3 
6-4 

•92 
766 
3447 
3)35-236 

11-745 

5-6 
3-2 

-  A      M 
4 

35-2 

8-0 
8-8 

=  A  +  4P  +  2QXg  =  J  area  of  Section  3. 

Vertical  Section  4. 
44 
2 
=  P                                          1F8  =  2Q 

=  4P 
=  A 

=  2Q                                                ,> 
=  A  -f  4P  +  2Q 

=  r 

r 

52-0 
•92 
"1040 
4680 

3)47-840 

15-946  = 

=  A-f4P  +  2QXo  =  i  area  of  Section  4. 

SHIP-BUILDING    AND    NAVAL   ARCHITECTURE.  469 

Vertical  Section  5. 

6-3  5-6  .  5-0 

^4  3^  2 

8-7  =  A        9-4  =  P  10-0  =  2  Q 

4 

37-6  =  4P 

8-7  =  A 

10-0  =  2Q 

56-3  =  A  +  4P  +  2Q 
•92  =  r 


1126 

5067 


3)51-796 


r 


17-265  =  A-f4P-f2QXo=}  area  of  Section  5. 

Half  areas  of  Vertical  Sections  1,  2,  3,  4,  and  5. 

No.  1 1-104  feet. 

2 6-256 

3 11-745. 

4 15-946 

5 17-265 

Displacement  of  the  body  under  the  fore  half-length  of  the  load- 
water  line  by  the  vertical  sections,  or  the  summation  of  the  vertical 
areas  1,  2,  3,  4,  and  5,  by  tlie  formula  for  the  solid,  as  being 
equal  to 

|  A'  -f  4  P'  +  2  Q'  \   x^  where  A'  =  sum  of  1st  and  5th  areas. 

P'  =  "      2d  and  4th  areas. 

Q'  =  «      3d  area. 

And  r'  —  distance  between  the  vertical  sections,  or  5-5  feet. 

1...  1-104            2...  6-256  3. ..11-745  =  Q' 

5. ..17-265            4. ..15-946  2 

18-369  =  A'   22-202  =  P'  23-490  =  2  Q' 

4 

8M08  =  4P' 
18-369  =  A' 
23-490  =  2  Q' 


130-667  =  A'  -f  4F  +  2Q' 
5-5  =  r' 


653335 
653335 

3)718-6685 


239-556    =A'  +  4P'+  2Q'  x  1-= cubic  ft.  of 

2     space  in  J  fore-body.  ^ 
479-112    =  cubic  feet  of  space  in  fore-body. 


4TO  THE    PRACTICAL   MODEL   CALCULATOR. 

Displacement  of  the  body  immersed  under  the  after  half-leitgth 
of  the  load-water  line  by  the  vertical  areas  5,  6,  7,  8,  and  9  of  the 
Table  of  ordinates. 

Vertical  Section  6. 

5,  as  fore  body.      6-1  5-5  4-6  =  Q 

17-265  W  JM  _2 

8-1  =  A     8-9  =  P  9-2  =  2Q 

_4 

3576  =  4P 
8-1  =  A 
9-2  =  2  Q 


•92 


1058 
4761 

3)48-668      _ 

16^222  =  A+4P+2Q  x^=  J  |  arfea  °,f 
3     \  Section  6. 

Vertical  Section  7. 

5-4  4-4  3-4  =  Q 

14  ^4  _2 

6-8  =  A     6-8  =  P  6-8  =  2Q 

_J* 
2T2  =  4  P 

6-8  =  2  Q 
6-8  =  A 

40-8  =  A  +  4P  +  2Q 
•92  =  r 
816 
3672 


3)3T536 


Vertical  Section  8. 

3-7  2-6  1-7  =  Q 

j6  JL-1  j2 

4:3  =  A"3:7  =  P  3-4  =  2Q 

^ 

14-8  =  4P 
4-3  =  A 
3-4  =  2  Q 

22-5  -A  +  4P  +  2Q 
j92==r 
450- 
2025 


3)2MOO 


SHIP-BUILDING    AND    NAVAL   ARCHITECTURE.  471 

Vertical  Section  9. 

•4  -35  -3  =  Q 

2  J25  _2 

•6  =  AT60='P  ~T6  =  2Q 

4 
2-4  =  4  P 

•6  =  A 
^6  =  2Q 

3-6  =  A  +  4P-f2Q 
•92  =  r 
~72 
324 


3)3-312 


1-104  =  A-f4P-f2Qxg  =  J  area  of  Section  9. 
Half  areas  of  the  vertical  sections  5,  6,  7,  8,  and  9. 

Sections.  Areas. 

5  .................................  17-265 

6  .................................  16-22 

7.,  ................................  12-512 

8.  ................................  6-9 

.   9  ................................  1-104 

Displacement  of  the  after-body  under  the  after  half-length  of  the 
load-water  line  by  the  vertical  sections,  or  the  summation  of  the 
immersed  areas  of  the  vertical  sections  5,  6,  7,  8,  and  9  by  the 
formula  for  the  solid  as  being  equal  to 


A'  +  4  P'  +  2  Q'  x  -g- 

where  A'  =  sum  of  the  5th  and  9th  areas. 
P'  =  "         6th  and  8th  areas. 

Q'  =  "         7th  area, 

and  rf  —  the  distance  between  the  vertical  sections,  or  5*5  ft. 

5. ..17-265  6. ..16-22  7. ..12-512  =  Q' 

9...  1-104  8...  6-900  2 

ET369  =  A'  2M20  =  P'  WWA  =  2  Q' 

4 

92-480  =  4  F 

25-024  =  2  Q' 

18-369  =  A' 


135-873  =  A'  +  4  F  +  2  Q' 

5^5  =  r 

679-365 
67-936 


3)747-3015 


249-1005  =A/-f4P'+2Q'x:_=cubicft. 

2       in  }  after-body.  6 

498-2010  After-body  in  cubic  ft.  of  space. 


472  THE    PRACTICAL    MODEL    CALCULATOR. 

Displacement  of  Fore-body  by  Horizontal  Sections. 

Horizontal  Section  V. 

0-4  6-0  5-0  =  Q 

6-3  3-0  __2 

(FT  =  A'  9-0  =  P  10-0  =  Q 

_  4 

5MO  =  4P 
10-00  =  2  Q 

6-70  =  A 
52T6  =  A  +  4P  +  2Q 

5*5  =  r 
2635 
2635 
3)289-85 


__ 

96-61  =  A  +  4  P  4-  2  Q  x     =  |  area  of  Section  1'. 

3 

Horizontal  Section  2'. 

•35  5-7  4-2  =  Q 

5-60  2-4  _2 

S7^  =  A          '  8-1  =  P  8-4  =  2  Q 

4 

324  =  4P 
8-4  =  2  Q 
5-95  =  A 

46-75  ==A  +  4P  +  2Q 
_  5-5  =  r 
"23375 
23375 
3)257-125 

85-708  =  A  +  4P  +  2Qx^  =  J  area  of  Section  2r. 

o 

Horizontal  Section  3'. 
.3  4.4  3-2  =  Q 

5-0  1-7 

^3  -  A  M  =  P  6-4  =  2  Q 

4 
244  =  4P 

5-3  =  A 
_64  =  2Q 
36-1  =  A  +  4P  +  2Q 

5-5  =  r 
"1805      ' 
1805 
3)198-55      _  _ 

66-18  =  A  +  4P  +  2Qx^  =  J  area  of  Section  3'. 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  473 

Horizontal  Section  4'. 
•25  3-2  2-2  =  Q 

3^_  I-Q  __2 

4-05  =  A  4-2  =  P  4-4  =  2  Q 

4 

16-8   =  4P 
4-05  =  A 
4-40  =  2  Q 

25-25  =  A  +  4P  +  2Q 
5-5  =  r 


12625 
12625 


3)138-875 

46-291  =  A  +  4P  +  2Q  x  J  =  { 

Horizontal  Section  5'. 
2-0  1-3  =  Q 


2-6  =  A  24  =  P 

_4 

9^6  =  4P 
2-6  =  A 
JM>  =  2  Q 

14:8  =  A 
5-5  =  r 


740 
740 


S 


area  of 


Displacement  of  the  fore-body  under  the  fore  half-length  of  the 
load-water  line  by  horizontal  sections,  or  the  summation  of  the 
horizontal  sections  of  the  fore-body  lr,  2',  3',  4r,  and  5r,  by  the 
formula  for  the  solid,  as  being  equal  to 


where  A'  =  sum  of  the  1'st  and  5rth  areas  ; 

Pr  =          u          2rd  and  47th  areas  ; 
Qr  =          "          3rd  area  ; 

and  r  =  the  distance  between  the  horizontal  sections,  or  *92  feet. 
Half  areas  of  the  Horizontal  Sections  T,  2;,  3',  4',  and  5'. 


1'  =  96-61. 

2'  =  85-708. 
3'  =  66-18. 


4r  =  46-29. 
5;  =  27-13. 


474  THE   PRACTICAL  MODEL    CALCULATOR. 

Areas.  Areas.                                                            Areas. 

V...  96-61  2'...  85-708                         3'.. .66-18  =  Q' 

5'.. .27-13  4'.  ..46-290                                      2 


123-74  =  A'        131-998  =  F  132-36  =  2  Q' 


527-992  =  4  F 
123-740  =  A' 
132-360  =  2  Q' 

784-092  =  A'  +  4F  +  2Q' 
•92  =  r 

1568184 

7056828 


3)721-36464 

ft- in 


^~EV  *4t)  — —  Xi.    ~T~TCJL      ~t~  *J  V^     /S  o  *\      1  i"  1_       J 

3      \  Jfore-body. 

480-90       =  fore-body  by  horizontal  sections  in 
cubic  feet  of  space. 

Displacement,  by  horizontal  sections  of  the  body  immersed  under 
the  after  half-length  of  the  load-water  line,  or  by  the  horizontal 
areas  1',  2',  3',  4',  and  5',  of  the  table  of  ordinates. 

Calculated  areas  of  V,  2/  3',  4',  and  5'. 

Section  V  After-body. 

6-3  6-1  5-4  =  Q 

•4  3-7  2 

6-7  =  A  9-8  =  P  10-8  =  2Q 

4 

39-2  =  4  P 

10-8  =  2  Q 

6-7  =  A 

56^=  A  +  4P  +  2Q 
5-5  =  r  ^ 

2835 

2835 

3)311-85 

103*95  A    ]    /iT>_i_orkv j    2  area  ot 


SHIP-BUILDING    AND   NAVAL   ARCHITECTURE.  475 

Section  2'  After-body. 

5-6  5-5  4-4 

_13J>  2-6  _2_ 

fr95  =  A      8T=P  8-8  =  2Q 


32-40  =  4P 

5-95  =  A 

8-80  =  2Q 

iTOUS  =  A  +  4P  +  2Q 
5-5  =  r 


23575 
23575 
3)259-325 


86-441  =  A  +  4P  +  2Qx-g=J  area  of  Section  2'. 

Section  3'  After-body. 
5-0  4-6  3-4  =  Q 

Q  1 .T  »O 

O  J-    I  ~ 

51$  =  A       IKS  =  P  6:8  =  2Q 

_4 

2^2  =  4P 
5-3  =  A 

6-8  =  2  Q 

37-3  =  A  +  4P  +  2Q 
5-5  =  r 

~1865 
1865 
3)205-15       , 


68-38  =  A  +  4P+2Qx-g=J  area  of  Section  3'. 

Section  4'  After-body. 
3-8  3-4  2-4  =  Q 

•25  1-1  _2 

^05  =  A      4-5  =  P  4-8  =  2Q 

_  4 

18-00  =  4P 
4-05  =  A 
4-80  =  2Q 

26-85  =  A  +  4P  +  2Q 
_  5^>  =  r' 

13425 
13425 


3)147-675       _       , 

49-225  =  A  +  4P4-2QXg-  =  -|  area  of  Section  4'. 


476  THE    PRACTICAL    MODEL    CALCULATOR. 

Section  5'  'After-body. 

2-4                 2-0  1-4  =  Q 

j2                _*6  2 

2T6  =  A2T6  =  P  2T8"=2Q 


10-4  =  4P 

2-8  =  2  Q 
2-6  =  A 
15-8  =  A 
5-5  =  r' 


790 
790 
3)86-90 


28-96  =  A  +  4P  +  2Qxg-  =  J  area  of  Section  5'. 

Displacement  by  horizontal  sections  of  the  after-body  under  the 
after  half-length  of  the  load-water  line,  or  the  summation  of  the 
horizontal  sections  of  the  after-body,  1',  2',  3',  4',  and  5',  by  the 
formula  of  the  solid,  as  being  equal  to 

A'  +  4  F  +  2  Q'  x  |p 
Half  areas  of  the  After  Horizontal  Sections,  V,  2',  3',  4',  and  5'. 

Sections.  Areas. 

V 103-95. 

2' 86-44. 

3' 68-38. 

4' 49-22. 

5'..,  .  28-96. 


1'... 103-95  2'. ..86-44  3'...  68-38  =  Q' 

5'...  28-96  4;... 49-22  2 

132-91  =  A'      135-66  =  F  136-76  =  2 

4 

542-64  =  4P' 

132-91  =  Ar 

136-76  =  2Q' 

812-31  =  A'  -f  4F  +  2Q' 

-92  =  r 
162462 
731079 


3)747-3252      r 

-  20'  x  "3  =  cubic  ft.  of 


2     J  after-body  by  horizontal  sections. 
498-2168  =  After-body  by  horizontal  sections 
in  cubic  feet  of  space. 


SHIP-BUILDING   AND    NAVAL    ARCHITECTURE.  477 

DISPLACEMENT. 

By  Vertical  Sections.  By  Horizontal  Sections. 

Cubic  Feet.  Cubic  Feet. 

Fore-body   (p.  469)   479-11       Fore-body   (p.  474)   480-900 
After-body  (p.  471)   498-20       After-body  (p.  476)   498-216 

Sum   977-30  Sum   979-116 

Half  488-65  Half  489-558 

Cubic  Feet. 

By  Horizontal  Sections 979-116 

By  Vertical  Sections 977-300 

Difference 1-816  cubic  feet. 

Cubic  Feet.       'v    . 

979-49  =  capacity  or  displacement  in  cubic  feet  of  space. 

The  mean  weight  of  salt  and  fresh  water  gives  35  cubic  feet  of 
space,  when  filled  with  water,  to  be  equivalent  to  a  ton  avoirdupois ; 
thence  the  displacement  in  cubic  feet  of  space  being  divided  by  35 
will  give  the  weight  of  the  volume  displaced  in  tons  avoirdupois ; 
or  979-49  being  divided  by  35  gives 

5)979-49 
7)195-898 

27-985  tons,  the  weight  of  the  calculated 
immersed  body  in  tons. 

AREA    OF    THE    MIDSHIP    SECTION,    OR    OP    THE    GREATEST    TRANSVERSE 

SECTION. 

Section  at  5. 

1-1. ..6-3      2-2. ..6-0  3-3. ..4-8  =  Q 

5-5.  ..j2      4-4. ..2-3  _2 

6-5  =  A      8-3  =  P  9-6  =  2  Q 

_4 

33-2  =  4  P 
6-5  =  A 

9-6  =  2  Q 

49-3  =  A  +  4P  +  2Q 

r 
1-25  =  o  where  r  —  the  depth,  from  1  to  5,  di- 

oTTT  vided  by  4  =  5  ft.  by  4  = 

92865          m^ 

493 
3)61-625 

SIO^I  =  A  +  4P  +  2Q  x  ^  =  /  2  area  of  mid- 
2  \  ship  section. 

41-082  =  Area  of  midship  section  without  keel. 


478  THE    PRACTICAL    MODEL    CALCULATOR. 

LOAD- WATER    LINE. 

Area  of  the  load-water  line,  or  area  of  the  assumed  deepest  plane 
of  immersion,  delineated  on  the  half-breadth  plan,  and  marked  by 
the  curve  AB.  From  the  table  of  ordinates,  p.  467,  we  have 

•4  3-0  5-0 

•4  6-0  6-3 

•8  =  A         ||'l  ^ 

-£l  16-7  =  Q 

18-8  -  P  _2 

— -  33-4  =  2  Q 

75-2  =  4  P 

•8  =  A 
33-4  =  2  Q 


109-4 
5-5 

5470 
5470 


3 )  601-70 

200^6  =  A  +  4P  +  2Q  x  -  =  /  2  area  of  load- 

3        |^  water  line. 
200-56  —  J  area  of  load-water  section  in  superficial  feet. 

9 


401*12  =  area  of  load-water  section,  which  amount  of  area  being 
divided  by  12,  will  give  the  number  of  cubic  feet  of  space  that  would 
be  contained  in  a  zone  of  that  area  of  ail  inch  in  depth,  and  that 
result  being  again  divided  by  35,  as  the  number  of  cubic  feet  of 
water  equivalent  to  a  ton  in  weight,  will  give  the  number  of  tons 
that  will  immerse  the  vessel  one  inch  at  that  line  of  immersion. 
12)401-12  =  area  of  load-water  section  in  superficial  feet. 
5 )  33-42  =  cubic  feet  in  zone  of  one  inch  in  depth. 
77^684 

-955  =  tons  to  the  inch  of  immersion  at  load-water  line. 

CENTRE   OP   GRAVITY    OF    THE    DISPLACEMENT. 

Estimated  from  Section  1,  considered  as  the  Initial  Plane. 

Distinguishing  J£  Vertical 

No.  of  the  Areas.  Areas.  Moments. 

1 1-104  x  0 000-000 

2 6-256  x  1 6-256 

3 11-745  x  2 23-490 

4 16-069  x  3 48-207 

5 17-265  x4 69-060 

6 16-222  x  5 81-110 

7 12-512  x  6 75-072 

8 6-900  x  7 48-300 

9...   ,  1-104  x  8...  8-832 


SHIP-BUILDING    AND    NAVAL    ARCHITECTURE.  479 

Moments  placed  in  the  -Rule. 


Sum  =A  +  4P-f2Qx-g 

000-000  6-256  23-490 

8-832  48-207  69-060 

T832  -  A  81'110  75'072 


48-300 


167-622 


183-873  =  P      2 

335-244  =  2  Q 
735-492  =  4P 

8-832  =  A 
335-244  =  2Q 

1079-568  =  A  +  4P  +  2Q 
5-5  =  rr 

5397840 
5397840 

3 )  5937-6240 

1979-208    =  A  +  4P  +  2Q  x  rL  = 
t  o 

sum  of  the  moments  of  half  the  displacement  from  section  1,  in  in- 
tervals of  space  of  5-5  ft. ;  and  the  half  displacement  in  cubic  feet 
by  vertical  sections  is  488-650  (p.  477)  cubic  ft. ;  whence  it  is 
found,  by  dividing  the  moment  1979-208  by  488-650,  that  the  dis- 
tance of  the  centre  of  gravity  of  displacement  from  the  section  1 
is  as  follows  : — 

488-65 )  1979-208  ( 4-05  intervals  from  1. 
195460         interval  =  5-5  ft. 

246080 
244325 

1755  therefore  4-05  X  5-5  =  22-27  ft.  = 
distance  of  the  centre  of  gravity 
of  the  calculated  immersed  body 
\  v^  from  1. 

DEPTH    OF    THE    CENTRE    OF    GRAVITY    OF    THE    DISPLACEMENT    BELOW 
THE   LOAD-WATER   SECTION. 

Fore-body.     After-body. 

Section.  Areas.  Areas.  Sum  of  the  Areas.  Moments. 

96-61          f  103-95 200-56  x  0  =  000-000 

85-708  g       86-44 172-148  x  1  =  172-148 

66-18  i\    68-38 134-56  x  2  =  269-12 

46-29  49-22 95-51  x  3  =  286-53 

27-13  H  [  28-96 56-09  x  4  =  224-36 


480 


THE  PRACTICAL  MODEL  CALCULATOR. 


000-00 
224-36 

224-36 


269-12 

9. 


172-148 

286-530 

458-678 
4 

1834-712  =  4  P 
224-360  =  A 
538-240  =  2  Q 

2597-312  =  A  +  4P 
•92  =r 


Q 


P   538-24  =  2  Q 


2Q 


5194624 

23375808 

3)2389-52704 
796-509 


A  +  4P  -f  2Q  x  '  = 


sum  of  the  moments  of  the  half  displacement  calculated  from  the 
load-water  line :  the  half  displacement  by  horizontal  sections  is 
489-588  (p.  477)  cubic  feet ;  the  sum  of  the  moments  of  the  half 
displacement  796-509  ft.,  being  divided  by  that  quantity,  will  give 
the  distance  in  intervals  of  -92  ft. ;  the  centre  of  gravity  of  dis- 
placement is  below  the  load-water  line. 

Half  solid  of 
displacement.  Moments. 

489-558 )  796-509 )  1-62  intervals  of  -92  feet ;  therefore 


489558 

3069510 
2937348 

1321620 
979116 

342504 


1-62 
x  -92 

324 
1458 

1-4904  ft.  =  the  distance  the  cen- 
tre of  gravity  of  the  calcu- 
lated immersed  body  is  be- 
low the  load-water  section. 


DISTANCE   OF    THE    CENTRE    OF    GRAVITY    OF   THE    AREA    OF   THE    LOAD- 
WATER    SECTION    FROM    SECTION    1. 


No.  of  Section. 

Ordinates  of  Section  1 
from  the  Table,  p.  467. 

Distances  of  them  in 
intervals  of  5-5  ft. 
from  Section  1. 

Moments  ;  being  the  Pro- 
duct of  the  Areas  by  the 
respective  Distances. 

1 

•4 

~  o"~ 

000-00 

2 

3-0 

1 

3-0 

3 

5-0 

2 

10-0 

4 

6-0 

3 

18-0 

5 

6-3 

4 

25-2 

6 

6-1 

5 

30-5 

7 

5-4 

6 

32-4 

8 

3-7 

7 

25-9 

9 

•4 

8 

3-2 

SHIP-BUILDING    AND    NAVAL   ARCHITECTURE.  481 

The  moments,  for  summation,  put  into  the  rule. 
00-0  3-0  10-0 

3-2  18-0  25-2 

~8^2  =  A  30'5  32-4 

?^i                           67-6  =  Q 
77-4  =  P  2 

135-2  =  2  Q 


309-6  =  4  P 

3-2  =  A 
135-2  =  2Q 

448-0  =  A-f4P  +  2Q 
5-5  =  r1 

2240 
2240 

3)2464-0 

821-3  =  A  +  4  P  +  2  Q  x  t.  = 

3 

sum  of  the  moments  of  the  half  area  of  the  load-water  section 
reckoned  from  1 ;  the  half  area  of  the  load-water  section  is  200-56 
feet  (p.  478) ;  the  distance,  therefore,  of  the  centre  of  gravity  of 
the  load-water  section  from  1  will  be  found  in  intervals  of  space  of 
5-5  feet,  by  dividing  the  sum  of  these  moments  by  the  half  area, 
thiis:— 

Half  Area.  Moments.  No. 

200-56 )  821-3333  (4-09  intervals,  each 
80224  5-5  ft.  in  length. 

190933 
180504 
10429 

and  4-09  X  5-5  =  22-5  ft.  gives  the  distance  of  the  centre  of  gravity 
of  the  load-water  section  from  section  1  of  the  drawing. 

Relative  capacities  of  the  bodies  immersed  under  the  fore  and 
after  lengths  of  equal  division  of  the  load-water  line — 
By  former  calculations. 

After-body  immersed  contains 497*79  cubic  ft.  of  space, 

Fore-body          "  "       481-70  cubic  ft.  of  space. 

Difference. 16-09  = 

the  excess  in  cubic  feet  of  space  of  the  body  displaced  under  the 
after  half-length  of  the  load-water  line  over  that  under  the  fore- 
half  of  the  same  line — 

Sum  of  the  bodies  (by  former  calculation)  or  whole  1  070.40 

displacement  in  cubic  feet  of  space  (p.  477) / 

equal  to  9-7949  hundreds  of  cubic  feet  of  space,  whence  16-09,  or 
the  difference  between  the  two  bodies  in  cubic  feet,  being  divided 
by  9-7949,  or  the  displacement  expressed  in  terms  of  the  hundreds 

31 


482  THE   PRACTICAL   MODEL   CALCULATOR. 

of  cubic  feet  of  space,  will  give  the  excess  for  every  hundred  cubic 
feet  of  the  whole  displacement. 

Displacement  in  Excess  in 

Hundreds  of  Cubic         Cubic  Feet 

Feet  of  Space.  of  Space. 

9-7949  )  16-09000  (  1-6  =  Ratio  of  the  excess  of 

97949  the  after-body  of  dis- 

629510  placement     over     the 

587694  fore-body  of  the  same, 

•41816  denoted  by  a  per-cent- 

age  of  the  whole  dis- 

METACENTRE.         placement. 

A  measure  of  the  comparative  stability  of  a  ship,  or  the  height 
of  the  metacentre  above  the  centre  of  gravity  of  displacement  esti- 

7V    (i  X 

mated,  from  the  expression  §  /  —  —  ,  in  which  /  is  the  sign  of  in- 
tegration and  signifies  sum  :  — 

y    =  the  ordinates  of  the  half-breadth  load-water  section. 

dx  =  the  differential  increment  of  the  length  of  load-water  section. 

D  =  displacement  of  the  immersed  portion  of  the  body  in  cu- 
bic feet  of  space. 

Ordinates  from  the  table.  Cubes  of  the  Ordinates. 

•4  ....................................  00-064 

3-0  ....................................  27-000 

5-0  ....................................  125-000 

6-0  ....................................  216-000 

6-3  ....................................  250-047 

6-1  ....................................  226-981 

5-4  ....................................  157-464 

3-7  ....................................  50-653 

•4  ....................................     0-064 

Cubes  placed  in  O'Neill's  rule  for  summation  of 


Area  =  (A  +  4P  +  2  Q)  X 


00-064 
00-064 


••  A 

3) 
f  ?/3  dx  = 

27-000 
216-000 
226-981 
50-653 

=  P 

=  4P 

=  2Q 
=  A 

=  r' 

125-000 
250-047 
157-464 

532-511  =  Q 

2 

520-634 
4 

1065-022  •=  2  Q 

r' 

2082-536 
1065-022 
000-128 

3147-686 
5-5 

15738430 
15738430 

17312-2730 

=  5770-7576 

=  A  -f 

4P  +  2Q  x  £*- 

0 

SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  483 

summation  of  the  cubes  of  the  ordinates  of  the  load-water  section  ; 
and  the  height  of  the  metacentre  above  the  centre  of  gravity  of 

y3  dx 
displacement  is  expressed  by  f  /         ,  in  which  expression  yz  dx  = 


5770-75  and  D  =  979-1  (p.  477)  whence  f  x  -j-  =  3-98  feet 

is  the  height  of  the  metacentre  above  the  centre  of  gravity  of  the 
displacement. 

RESULTS   OF   THE   CALCULATIONS. 

1st  Method. 

Displacement  in  cubic  feet  of  space     =  979-149. 
Displacement  in  tons   of  35  cubic  )          07.  Q7  i 

(•  r>  ,  f  —       ~.  [  '<J  (  **, 

feet  ot  water  to  a  ton  ...............  J 

Area  of  midship  section  ...............     =    41-08  superficial  feet. 

Area  of  load-water  line  or  plane  at  )        ,,  ni  -,  0  G  .  -,  e 

the  proposed  deepest  immersion..  }=  401'12  s»Perfi«'al  *»*. 
Tons  to  one  inch  of  immersion  at  "I        Qrr  , 

thatflotation  ..........................  }=    '955  tons' 

Longitudinal  distance  of  the  centre  1 

of  gravity  of  displacement  from  >  =  22-22  feet. 

section  1.  J 

Depth  of  the  centre  of  gravity  oH 

displacement  below  the  load-water  >=  1-4904  feet. 

section  .................................  J 

Distance  of  the  centre  of  gravity  of  ^ 

the  load-water  section  from  verti-  >  =  22-5  feet. 

cal  section  1  ..........................  J 

Relative  capacity  of  the  after-body  ^ 

in  excess  of  the  fore-body  in  cubic  >  =  16-09 

feet  of  space  ..........................  J 

Per-centage  on  the  whole  displace-  \  __  •*  /w. 

ment  ...................................  /  = 

Height  of  the  metacentre  above  the') 

centre  of  gravity  of  displacement,  j 

estimated    from    the    expression  ^  =  3-98  feet. 


3          D     ' 

The  young  naval  architect  has  thus  been  led  through  the  essen- 
tial calculations  on  the  immersed  portion  of  a  ship  considered  as  a 
floating  body.  The  term  essential  has  here  been  used  under  a 
knowledge  that  the  table  of  results  might  have  been  swollen  to  a 
small  volume  by  a  lengthened  comparison  of  the  elements  of  the 
naval  construction,  such  as  the  ratio  of  the  area  of  the  midship  sec- 
tion to  the  area  of  the  load-water  section,  and  that  of  the  area  of 
the  midship  section  to  the  circumscribing  parallelogram  ;  data  that 
will  always  suggest  themselves  to  the  mind,  and  furnish  salutary 
exercise  for  his  judgment,  while  the  introduction  of  such  com- 
parisons into  these  rudiments  might  deter  the  novice  from  entering 


484 


THE  PRACTICAL  MODEL  CALCULATOR. 


A 

2 

*    2P 

P 

§ 
PH 

g   «» 

2P 

A 
"2 

B 
C 

jljg  § 

0 

1 

CO 

? 

0 

2 

Hi* 

. 

"H 

« 

i—  i 

lip  i 

j 

1—  1 

CO 

94 

? 

Functions  of  Longitudinal  Areas. 

ill 

8 

CD 

8 

o 

1—  1. 

1 

CO 
CO 

(N 

T-H 

o 

i—  i 

0, 

CO 

1 

0 
0 

8 

CD 

§ 

W3 

r—  i 

T—l 

co 

1 

13-17 

•4 

0-20 

3-0 

1-50 

5-0 

2-50 

6-0  |  3-00 

6-3 

3-15 

0 

1 

o 

1 

1 

CO 

46-34 

•35 

0-70 

2-4 

4-80 

4-2 

8-40 

5-6 

11-20 

5-0 

11-20 

i 

i 

CO 

0 
CN 

CO 

CO 

CNI 

CO 

18.05 

•3 

0-30 

1-7 

1-70 

3-2 

3-20 

4-4 

4-40 

I 
5-0  |  5-00 

0 

SN 

?         ° 

CM 

o 

CD 

O 

1 

25-24 

•25 

0-50 

1-0 

2-00 

2-2 

4-40 

3-2 

6-40 

3-8 

7-60 

o 
o 

6 

0 
CO 

8 

0 
CN 

0 

Tf! 

3-70 

•2 

0-10 

•4 

•20 

1-3 

•65 

2-0 

1-00 

2-4 

1-20 

i 

Functions 
of  Vertical 
Areas. 

1-80 

10-20 

19-15 

26-00 

28-15 

Function  of 
the  Solid  by 
Longitudinal 
Areas  10C-50. 

Multipliers     . 
for  Solid.    ^ 

c< 

t-H 

« 

H(M 

lillt  ! 

s 

T-H 

05 
i-H 

8 

CSI 

1 

o 

T-H 

c, 

CO 

« 

Function  of  the") 
Solid  by  Ver-  }- 
tical  Areas  J 

Hi! 

o 

I 

8 

rH 

CO 

SHIP-BUILDING   AND   NAVAL   ARCHITECTURE.  485 


AFTER-BODY. 

ls=  ls5 

O  W^l>  >-d  &  ITJ  ti||> 

1* 

1 

i 

Si 

1 

8 

11 

i 

- 

| 

- 

^ 

Jllfl! 

1 

i 

| 

s 

8 

3-glSJj       _, 

3&  &  f  d  ••? 

JJIF3 

s 

1 

gj 

1 

„ 

0 

«0 

^ 

00 

2 

CO 

1 

! 

0 

i 

0 

i 

14-7 

Sum  of 
the  Func- 
tions of 
Fore 
and  After- 
bodies. 

Moments 
for  the 
Centre 
of  Gra- 
vty  of 
Dis- 
place- 
ment. 

oiooo-oo 

J-3 

3-15 

6-1 

3-05 

5-4 

2-70 

3-7 

•4 

0-20 

1 

§ 

s 

R 
£ 

s 

0 

gj 

2 

47'14 
18-65 
26-84 
S-95 

i 

1    93-48 

5-6 

11-20 

5-5 

11-00      |  4-4 

8-80 

2-6 

5-20 

•35 

•70 

I 

2 

? 

w 

s 

ss 

9 

f 

1 

2 

2   73-40 

VO 

5-00 

4-6 

4-60 

34 

3-40 

1-7 

1-70 

•3 

0-30 

2 

i 

0 

\ 

g 

o 

SSI— 

1 
i 

3  156-24 

3-8 

7-6D 

3-4 

6-80 

2-4 

4-80 

1-1 

2-20 

•25 

0-50 

§ 

3 

! 

1 

o 
o 

SSI  «• 

4   30-60 

1-4 

1-20 

2-0 

MM 

1-4 

•70 

•6 

•30 

•2 

•10 

Functions 
of  Vertical 
Areas. 

23-15 

26-45 

20-40 

11-25 

1-80 

110-75             217-25     353-72 

1 

Multiplier3\« 
for  Solid.  -x 

« 

- 

« 

'» 

§  «  S3S  o 

I 

I 

1 

? 

; 

• 

o 

- 

00 

! 

9? 

i 

i 

fe 

1 

486  THE  PRACTICAL  MODEL  CALCULATOR. 

on  a  task  that  would  thence  seem  to  be  involved  in  such  voluminous 
results.  For  the  second  method  of  calculation,  the  table  of  ordi- 
nates is  in  two  portions,  viz.  the  fore  and  after-bodies  under  the 
division  of  the  load-water  section  into  two  equal  parts,  the  length 
of  such  section  being  restricted  to  the  distance  from  the  fore-edge 
of  the  rabbet  of  the  stem  to  the  after-edge  of  the  rabbet  of  the 
post.  The  enlarged  tables  are  shown  at  pa,ges  484  and  485,  and 
the  directions  for  the  working  of  these  tables  have  been  given  at 
page  459,  observing  only  that  the  ordinates  have  not  been  herein 
inserted  in  red,  as  it  was  there  suggested,  to  insure  perspicuity  and 
accuracy. 

RESULTS   FROM   THE  TABLES. 

By  modified  rule.     Area  =  |'2+2]P  +  Q}lf 

And  solid  =  areas  for  ordinates  \  _  f  A'  Q'  1  -f-  — ' 

summed  by  rule  J  ~  (   2  ^  J         3 

Functions  of  the  areas  marked  B=<  £•  -f  2  P  -f  Q  > 

Function  of  the  solid  equal  to  B,  placed  in  O'Neill's  rules  = 
A'  -f  2  F  +  Q'  =  E 


Whence  displacement  =  E  X  -Q-  x  -g-,  in  the  example  r  =  '92 

r'  =  5-5. 

2r      2r'  1-84      11 

Therefore  J  displacement  =  E  X  -g-  X  -g-  =  E  X  — g-  X  -g-  = 

20-24 
E  x  — <p. 

VALUE   OF   E   FROM   THE   TABLES   BY   VERTICAL    SECTIONS. 

Table  1...  106-50  =  submultiple  of  the  fore-body  by  vertical  sections. 
Table  2... 110-77=          "  after-body          " 

217-27=  sum  of  the  submultiples  =  E. 

20-24      217-27  x  20-24 
\  displacement  =  E  X  — g—  =  -      — g—      -  =  24-14  x  20-24  = 

488-5936  =  \  solid  of  displacement  by  the  summation  of  the 

2  vertical  areas  given  in  cubic  feet  of  space. 

5)977-1872 
7)195-4374 

27*92      =  Displacement  by  vertical  sections  in  tons  of  35 
cubic  feet  of  space. 

VALUE   OF   E   FROM   THE   TABLES   BY   HORIZONTAL    SECTIONS. 

Table  1...  106-50  =  submultiple  of  the  fore-body  by  horizontal 

sections. 
Table  2... 110-75  =  submultiple  of  the  after-body  by  horizontal 

sections. 
From  whence  the  same  results  will  be  obtained. 


SHIP-BUILDING  AND   NAVAL  ARCHITECTURE.  487 

AREA   OF   MIDSHIP   SECTION. 

From  table  1...28-15  =  submultiple  of  the  area  of  Section  5. 
1-84  =  2  r 


11260 
22520 
2015 

3)51-7960 

17-265    =  J  area  of  upper  space  of  midship  section. 
3-275    =  J  area  of  the  lower     "       "     below  d  d, 

=  }  area  of  midship  section. 


41-08    =  area  of  midship  section. 

AREA   OP   THE   LOAD-WATER  LINE. 

From  table  1... 26-35  =  submultiple  of  the  area  of  the  fore-body. 
From  table  2. ..28-35  =          "  "  after-body. 

54-70  =  submultiple  for  J  area  of  load-water  line. 
11  =  2  r' 


3)601-7 


200-56  =  J  area  =  -2 
2 


12)401-12  =  area  of  load-water  line. 
5)33-42 

7)6-684 

•955  =  tons  per  inch  of  immersion  at  the  load- 
water  line. 

POSITION    OF   THE   CENTRE   OF   GRAVITY   OF   DISPLACEMENT. 

By  table  2. ..878-86  =  moments  from  Section  1. 

and  E 217-27  =  corresponding  function  of  the  displacement. 

217-27)  878-86  (-404  intervals  of  5-5  feet,  giving  4-04  X 
869-08  5-5  =  22-22  feet  as  the  distance 

of  the  centre  of  gravity  of  the  dis- 
97800  placement  from  Section  1. 

86908 

10892 


488  THE   PRACTICAL  MODEL   CALCULATOR. 

DEPTH  OP  THE   CENTRE  OF  GRAVITY   OP  THE  DISPLACEMENT   BELOW  THE 
LOAD-WATER   SECTION. 

By  table  2.. .353-72  =  moments  from  load-water  line. 

and  E 217*25  =  corresponding  function  of  the  displacement. 

217-25)  353-72  (1-62  intervals  of  -92  feet,  giving  1-62  x 
217-25  -92  =  1-4904  as  the  distance  that 

136*470  ^e  cen^re  °f  gravity  of  displace- 

130*350  ment  is  below  the  load-water  line. 

61200 
43450 


17750 

POSITION    OF    THE   CENTRE    OF   GRAVITY  OF   THE    LOAD-WATER    LINE   OF 
DEEPEST    IMMERSION. 

From 'table  1 26-35  ft.     From  table  2. ..224-000  =  moments 

2 28*35  from  1st  section. 

Function  for  area..54-7)  224*0  (4-09  intervals  of  5*5  feet,   giving 

218-8  4-09  x  5*5  =  22-495  feet 

5200  as   *ke    distance   that   the 

4923  centre   of   gravity   of  the 

load-water  section  is  from 

•277  vertical  section  1. 

RELATIVE    CAPACITIES    OF    THE    CALCULATED    IMMERSED    BODIES    CON- 
TAINED  UNDER   THE   FORE   AND    AFTER-LENGTHS    OF   EQUAL   DIVISION 

OF   THE   LOAD-WATER   LINE. 

Feet. 

From  table  1... Function  for  the  fore-solid 106*50 

From  table  2... Function  for  the  after-solid 110-75 


4-25 
Sum  of  the  functions 217*25 

The  difference,  4*25  feet,  expresses  the  excess  in  cubic  feet  of  space 
of  the  body,  displaced  under  the  after  half-length  of  the  load-water 
line,  over  that  under  the  fore  half-length  of  the  same  line,  and  the 
sum  of  the  functions,  217*25,  is  equal  to  2*1725  hundreds  of  cubic 
feet  of  space ;  whence,  4*25  feet,  or  the  difference  between  the 
functions  for  the  two  bodies,  being  divided  by  the  function  2-1725, 
or  the  function  for  the  displacement  of  the  calculated  body  ex- 
pressed in  terms  of  hundreds  of  cubic  feet  of  space,  will  give  the 
excess  for  every  hundred  cubic  feet  of  that  displacement : 

Function  of        Excess  in 

Displace-        Cubic  Feet 

ment.  of  Space. 

2*1725 )  4*25000  ( 1*9  ratio  of  the  excess  of  the  after- 
2*1725  body  of  calculation  over  the 

2077  50  fore-body    of  the  same,   de- 

-i  o££0£  noted  by  a  per-centaare  of  the 

xyooznj  i*    i  i     i   .    i    i 

displacement   calculated   by 

•12225  the  table  of  ordinates. 


SHIP-BUILDING   AND   NAVAL   ARCHITECTURE.  489 

HEIGHT    OF    THE    METACENTRE    ABOVE    THE    CENTRE    OF    GRAVITY    OF 

DISPLACEMENT. 

From  table  2. ..The  summation  of  the  functions^ 

of  the  cubes  of  the  ordinates  for  the  value  of  V  =  1573*843. 

The  corresponding  function  for  the  solid =    217*25. 

from  whence  the  height  of  the  metacentre  above  the  centre  of 

gravity  of  displacement,  expressed  by  •$  /  is  as  follows : 

=  1573-843  x  -4^  where  rf  =  5-5  feet  = 
=  5770-75  feet. 


2r      2r' 
(Page  485)  217-27  x  -«   x  -g-  =  J  displacement  =  488-5936  feet, 

whence  displacement  or  D  =  977-1872 ;  and  thence 
2    y3dx  _  2       5770-75         11541-53 
3^    D      "  3  x  97771872  ~  2931-5616  = 

RESULTS   OBTAINED   UNDER   THE   TWO    METHODS   OF   CALCULATION 
CONTRASTED. 

Old  Method.  Second  Method. 

Displacement  in  cubic  feet  of  space...       979-139  977-187 

Displacement  in  tons  of  35  cubic  feet 

of  water  to  a  ton 27-985  27*92 

Superficial  ft.  Superficial  ft. 

Area  of  midship  section 41-08  41-08 

Area  of  load-water  line  or  plane  at 

the  proposed  deepest  immersion 401*12  401*12 

Tons  to  one  inch  of  immersion  at  line 

of  notation *9526  tons.  *955tons. 

Longitudinal  distance  of  the  centre  of 
gravity  of  the  displacement  from 
section  1 22-22  ft.  22-22  ft. 

Depth  of  the  centre  of  gravity  of  dis- 
placement below  the  load-water  sec- 
tion   1-4812  ft.  l-4904ft. 

Relative  capacities  of  the  bodies 1*6  per  cent.      l-9perct. 

Height  of  the  metacentre  above  the 

centre  of  gravity  of  displacement...  3-98  ft.  3-98  ft. 

THIRD   METHOD   OF   CALCULATION. 

CALCULATIONS  ON  THE  DRAUGHT  OF   THE   YACHT  OF  36    TONS  USING   THE 
CURVE    OF    SECTIONAL  AREAS. 

The  load-water  line  AB,  in  the  sheer  plan,  is  divided  into 
two  equal  parts  at  the  point  C,  and  those  equal  parts  are  again 
subdivided  at  the  points  D  and  E ;  at  the  points  C,  D,  and  E, 


490 


THE  PRACTICAL  MODEL  CALCULATOR. 


t, 


8-9 


Ordinates. 


RH  =  2-4  feet. 
QI    =  4-1    " 
PK  =  2-45  " 


DN  =  5-8  feet. 
CM  =  5-0    " 
EO  ==  4-2    «' 

AB  =  44  feet. 
FG  =  44     » 
FI   =  22     " 

10 


22      feet. 


QG  =  22-37 
FQ  =  22-37 


SHIP-BUILDING   AND    NAVAL   ARCHITECTURE.  491 

thus  obtained,  the  transverse  vertical  sections  of  the  vessel'  are 
delineated. 

The  length  of  the  load-water  line  from  the  fore  edge  of  the  rab- 
bet of  the  stem  B,  to  the  after  edge  of  the  rabbet  of  the  post  A,  is 
next  drawn  below  and  parallel  to  the  base  line  SF  of  the  sheer  plan  ; 
this  line,  FG,  becomes  the  base  line  of  the  curve  of  the  sectional 
areas.  The  common  sections  of  the  transverse  vertical  sections 
of  C,  D,  and  E,  (which  will  be  straight  lines,)  with  this  horizontal 
and  longitudinal  plan,  are  drawn  from  their  respective  points  of 
division,  H,  I,  and  K,  in  half-breadth  plan.  The  areas  of  these 
transverse  vertical  sections  at  D,  C,  and  E,  are  then  calculated, 
as  before,  thus  :  — 

Area=JA  +  4P  +  2QJx  g={^  +  2P  +  Q  1  x  ^;or, 
Area-  1  A  +  2P  +  3  Q  1  xfr-j^  +  P  +  1-5Q  }  X  |  r. 

Half  Area  of  Transverse  Vertical  Section,  at  (7,  ly  Rule  1, 
or,  }  Area 


1st.  ...6-3  2d  ...6-0  3d..  .4-8  =*  Q 

Last...   -2  4th.  ..2-3 

2)6-5 
3-25  = 


24-65 -      +  2P  +  Q 


7395 
19720        

A  2r 

20-4595  =-2^-2P  +  Qx-g-=J  area 

of  section  C  in  feet. 

CM         5-0 
CM,  or  depth  =  5-0  feet,  whence  -T-,  or  -r-  =  1'25  =  r  = 

2r      2x1-25      2-5 
distance  between  the  ordmates,  and  -«-  = o =  -Q-  =  '83  feet. 


492  THE   PRACTICAL    MODEL   CALCULATOR. 

Half  Area  of  Section  (7,  ly  Rule  2, 

or,  J  area  =|^|  +  P  +  1-5  Q  j  x  |  r. 

1st.  ...6-3  P  =  0  5-6    2d. 

Last..._j2  3-05  3d. 

2)6-5  1HJ5  =  Q. 

"3^25  =  -  4-32  =  i  Q. 

12-97  =  1-5  Q. 

l||}=4  +  P  +  l-5Q 

16-22 

5      =  3  r  =  CM  =  5-0  feet. 

4)81-10  

•  A  3 

20-275    =Jarea  =  ^  +  P  +  1'5  Q  X  j  r. 

Half  Area  of  the  Transverse  Vertical  Section  at  JEJ. 
1st.  ...5-0  2d.    ...4-2  3d.  ...2-9  =  Q 

Last..._^2  4th.  ...1-7 

2)"572  5-9  =  P 

-|  -l    Q    ___    O  T> 

11  o  —  ^  Jr 


2-9  =  Q 
17-3  =^  +  2P  +  Q 

EO       4-2 
EO,  or  depth  =  4-2  feet,  whence  -r-  =  -7—  =  1-05  =  r  =  dis- 

2r       1-05x2      2-1 
tance  between  the  ordmates,  and  -g-  = g =  —  =  .7  feet ; 

therefore, 

Area  =||  +  2P  +  Q|x^:  =  17-3  x  -7  =  12-11  =  half 
area  of  transverse  vertical  section  at  E. 

Half  Area  of  the  Transverse  Vertical  Section  at  D. 
1st.  ...5-40  2d.  ...3-5  3d.  ...1-46  =  Q 

Last...9^_  4th.. ..0-7 

2)5-6  4-2  =  P 

¥T- A       *    2 

"2  F4  =  2P 

2-8  -£ 

1-46  =  Q 
A 


12-66  =       +  2P  +  Q 


SHIP-BUILDING   AND   NAVAL   ARCHITECTURE.  493 

DN       5-8 
DN,  or  depth  =  5-8  feet,  whence  -j~  =  -j-  =  1-45  feet  = 

2r       2x1-45       2-9 
r  =  distance  between  the  ordmates,  and  -?r  =  -  o  --  =  ~o~  =* 

•97  feet  ;  therefore, 


Area  =2-2  +  P+Qx  =  12-66  X  -97  =  12-28  feet  = 
half  area  of  transverse  vertical  section  at  D. 

Half  Areas  of  the  Transverse  Vertical  Sections. 

Feet.  Feet. 

f  E  =  12-11^      Divided  by  5  as  the  depth  assumed  for  ("2-42 
At<  C  =  20-20  >the  zone,  give  the  ordmates  for  the  curves  4-04 

(D  =  12-28  j  of  sectional  areas,  as  ..........................  (2-45 

of  which  2-42  is  set  off  from  H  as  HR,  4-04  feet  from  I  as  IQ, 
and  2-45  feet  from  K  as  KP  ;  the  curve  IRQPG,  passing  through 
the  extremities  P,  Q,  and  R  of  the  ordinates  PK,  QI,  and  RH,  is 
the  curve  bounding  the  area  of  a  zone,  which,  to  the  depth  of  5  feet 
for  a  solid,  will  give  in  cubic  feet  of  space  the  half  displacement 
of  the  immersed  body,  or  the  displacement  of  the  yacht  to  the  line 
AB  of  proposed  deepest  immersion. 

To  measure  this  representative  area,  and  from  thence  the  solid, 
join  the  points  Q,  G,  and  I  by  the  straight  lines  QG,  QF,  dividing 
the  curvilinear  area  FRQPGF  into  the  two  triangles  QGI,  QFI, 
and  the  two  areas  GPQG,  FRQF.  The  triangles  by  construction 
are  equal,  and  the  area  of  each  one  of  them  is  equivalent  to 

-|  -  ,  or  the  whole  area  GQFIG  =  -  ^  -  x  2  =  GI  X  QI 

or  FI  X  IQ,»  FI  being  equal  to  IG,  each  being  the  half-length  of 
the  same  element,  the  load-water  line  or  line  of  deepest  immersion. 
The  areas  QPGQ,  QRFQ,  are  bounded  by  the  curve  lines  QPG, 
QRF,  which  are  assumed  as  portions  of  common  parabolas,  and 
under  such  an  assumption  their  respective  areas  are  equal  to  f  of 
the  circumscribing  parallelograms,  or  the  area  QPGQ  =  f  of 
GQ  x  a?,  and  the  area  FRQF  =  f  of  FQ  X  x'9  where  x  and  xr 
are  the  greatest  perpendiculars  that  can  be  drawn  from  the  bases 
QG  and  QF  to  meet  the  curves  QPG,  QRF. 

DISPLACEMENT. 

AB  by  a  scale  of  parts  =  44  feet,  whence  FI  or  IG  equal 
AB       44 
-g~  =  -g  feet  =  22  feet  ;  ordinate  QI  of  the  medial  section  = 

4-04  feet  ;  and  QG  =  FQ,  being  the  respective  hypothenuses  of  the 
equal  triangles  QGI,  QFI,  are  each  equal  to  N/IG2  +  QI2  = 

v/222  -f  I7^  =  v/484  +  16-32  =  ^500-32  =  22-37  feet  ;  and 
#,  by  measurement  with  a  scale  of  parts,  =  -6  foot,  and  x'  also 
•6  foot,  from  which  the  half  displacement  in  cubic  feet  of  space  will 
be  obtained  as  follows  :  — 


494  THE  PRACTICAL  MODEL  CALCULATOR. 

Area  FQGIF  =  GI  x  IQ.  cm*,  feet. 

Solid  under  the  1  =  GI  x  IQ  x  5  =  22   X  4-1   x  5  =     451-00 

cire<i  .c  v^^-*--*-  J 

Area  QPGQ  =  f  of  GQ  x  x 
Solid 

area 
Area  FRQF  =  f  of  FQ  X  x' 

44'74 
540-48 

or  area  of  the  triangle  QGI  +  area  of  the  triangle  QFI  -f  area 
of  the  space  QPGQ  +  area  of  the  space  FHQF  =  to  the  repre- 
sentative area  FRQPG,  which  being  multiplied  by  the  assumed 
depth  of  5  feet  for  the  zone  of  half  displacement  gives  540-48  cubic 
feet  of  space,  which  divided  by  35,  as  the  number  of  such  cubic 
feet  that  are  equivalent  to  one  ton  of  medium  water,  gives 

3 )  540-48 
7 )  108-09 

15-44  tons  for  half  displacement, 

and  that  the  whole  weight  of  the  body  is  equal  to  15-54  X  2  = 
30-88  tons  =  displacement  to  the  line  of  proposed  deepest  immer- 
sion AB. 

RELATIVE  CAPACITIES  OF  THE  BODIES  IMMERSED  UNDER  THE  FORE  AND 
AFTER  HALF-LENGTHS  OF  THE  LOAD-WATER  LINE,  AS  GIVEN  BY  THE 
DELINEATED  CURVE  OF  SECTIONAL  AREAS. 

The  triangles  QGI  and  QFI  being  equal,  the  relative  capacities 
of  the  fore  and  after-bodies  will  be  determined  by  the  proportion 
that  the  area  QPGI  bears  to  the  area  QRFI ;  and  as  these  areas 
involve  two  equal  terms,  or  that  the  base  FQ  =  the  base  QG,  it 
follows,  that  the  relation  of  these  areas  to  each  other  will  be  ex- 
pressed by  the  proportion  that  the  perpendiculars  x  and  x'  bear 
to  each  other.  In  the  example  given,  the  fore  and  after-bodies,  or 
the  displacements  under  the  fore  and  after  half-lengths  of  the  load- 
water  AB,  are  equal ;  as  the  perpendiculars  x  and  x'  taken  from 
the  diagram,  on  a  scale  of  equal  parts,  are  each  -6  of  a  foot. 

The  area  of  the  midship  section  is  denoted  relatively  by  the 
medial  ordinate  of  the  curve  of  sections  QI,  and  the  full  amount 
of  it  is  obtained  by  multiplying  the  function  QI  by  the  depth  of 
the  zone  M.  In  the  example : 

M  =  5 ;    QI  =  4-04  ;  then  half  area  of  medial  section  =  4-04  X  5 

5 


Area  of  midship  section 20*20 


TABLES   OF   LOGARITHMS. 


LOGARITHMS    OF    NUMBERS. 


N, 

Log. 

'rop. 
art. 

No. 

Log. 

'rop. 

JIUT.. 

No. 

Log. 

Prop.! 
i'art. 

No. 

Log. 

L'rop. 

L'art. 

1000 

000000 

1060  025306 

120 

049218 

1180 

071882 

1 

000434 

43 

1  !  025715 

41 

1 

049606 

39 

1 

072250 

37 

2 

000868 

86 

2  i  026124 

82 

2 

049993 

77 

2  072617 

73 

3 

001301 

30 

3  026533 

122 

3 

050380 

116 

3  072985 

110 

4 

001734 

73 

,  4 

026942 

163 

4 

050766 

154 

4  073352 

147 

5 

002166 

216 

5 

027350 

204 

5 

051152 

193 

5  073718 

183 

6 

002598 

259 

6 

027757 

245 

6 

051538 

232 

6 

074085 

2'20 

7 

003029 

303 

7 

028164 

286 

7 

051924 

270 

7 

074451 

256 

8 

003460 

346 

8 

028571 

326 

8 

052309 

309 

8 

074816 

293 

9 

003891 

389 

9 

028978 

367 

9 

052694 

347 

9 

075182 

330 

1010 

004321 

1070 

029384 

1130 

053078 

1190 

075547 

1 

004751 

43 

1 

029789 

40 

1 

053463 

38 

1 

675912 

36 

2 

005180 

86 

2 

030195 

81 

2 

053846 

77 

2 

076276 

73 

3 

00560(J 

128 

3 

030600 

121 

3 

054230 

115 

3 

076640 

109 

4 

006038 

171 

4 

031004 

162 

4 

054613 

153 

4 

077004 

145 

5 

006466 

214 

5 

031408 

202 

5 

054996 

191 

5 

077368 

181 

6 

006894 

257 

6 

031812 

242 

6 

055378 

230 

6 

077731 

218  I 

7 

007321 

300 

7 

032216 

283 

7 

055760 

268 

7 

078094 

254  1 

8 

007748 

343 

8 

032619 

323 

8 

056142 

306 

8 

078457 

290 

9 

008174 

385 

9 

033021 

364 

9 

056524 

345 

9  1  078819 

327 

1020 

008600 

1080 

033424 

1140 

056905 

1200 

079181 

1 

009026 

42  !    1 

033826 

40 

1 

057286 

38 

1 

079543 

36 

2  009451 

85  |    2 

034227 

80. 

2 

057666 

76 

2 

079904 

72 

3 

009876 

127 

3 

034628 

120 

3 

058046 

114 

3 

080266 

108 

4 

010300 

170 

4 

035029 

160 

4 

058426 

152 

4 

080626 

144  i 

5 

010724 

212 

5 

035430 

200 

5 

058805 

190 

5 

080987 

180 

0 

011147 

254 

6 

035830 

240 

6 

059185 

228 

6  I  081347 

216 

7 

011570 

297 

7 

036229 

280 

7 

059563 

266 

7  081707 

252 

8 

011993 

339 

8 

036629 

321 

8 

059942 

304 

8  |  082067 

288 

9  012415 

382 

9 

037028 

361 

9 

060320 

342 

9 

082426 

324 

1030  1  012837 

1090 

037426 

1150 

060698 

1210 

082785 

1 

013259 

42 

1 

037825 

40 

1 

061075 

38 

1 

083144 

36 

2 

013680 

84 

2 

038223 

79 

2 

061452 

75 

2 

083503 

71 

3 

014100 

126 

3 

038620 

119 

3 

061829 

113 

3 

083861 

107 

4 

014520 

168 

4 

039017 

159 

4 

062206 

160 

4 

084219 

143 

5 

014940 

210 

5 

039414 

198 

5 

062582 

188 

5 

084576 

179 

6 

015360 

252 

6 

039811 

238 

6 

062958 

226 

6 

084934 

214 

7 

015779 

294 

7 

040207 

278 

7 

063333 

263 

7 

085291 

250 

8 

016197 

336 

8 

040602 

318 

8 

063709 

301 

8 

085647 

286 

9 

016615 

378 

9 

040998 

357 

9 

064083 

338 

9 

086004 

822 

1040 

017033 

1100 

041393 

1160 

064458 

1220 

086360 

1 

017451 

42 

1 

041787 

39 

1 

064832 

37 

1 

086716 

35 

2 

017868 

83 

2 

042182 

7« 

2  065206 

75 

2 

087071 

71 

3 

018284 

125 

3 

042575 

118 

3  |065580  112 

3 

087426 

106 

4 

018700 

166 

4 

042969 

157 

4 

065953 

149 

4 

087781 

142 

5 

019116 

208 

5 

043362 

196 

5 

066326 

186 

5 

088136 

177 

6 

01953L 

250 

6 

043755 

236 

6  066699 

224 

6 

088490 

213 

7 

019947 

291 

7 

044148 

275 

7  067071 

261 

7 

088845 

248 

8 

020361 

333 

8 

044540 

314 

8 

067443 

298 

8 

089198 

284 

9 

020775 

374 

9 

044931 

354 

9 

067814 

336 

9 

089552 

319 

1050 

021189 

1110 

045323 

1170 

068186 

1230 

089905 

1 

02160J 

41 

1 

045714 

39 

1 

068557 

37 

1 

090258 

35 

2 

022016 

82 

2 

046105 

7£ 

2  |  068928 

74 

2 

090611 

70 

3  022428 

124 

3 

046495 

117 

3 

069298 

111 

3 

090963 

106 

4  022841 

165 

4 

046885 

156 

4 

069668 

148 

4 

091315 

141 

5  ;  02325L 

206 

5  |  047275 

190 

5 

070038 

185 

5 

091667 

176 

6  i  023664 

247 

6  1  047664 

234 

6 

070407 

222 

6 

092018 

211 

7  02407o 

288 

7  |  048053 

273 

7 

070776 

259 

7 

092370 

246 

8  i  024486 

330 

8  i  048442 

312 

8 

071145 

296 

8 

092721 

282 

9 

024896 

371 

9  048830 

35 

9 

071514 

333 

9 

093071 

317 

32 


LOGARITHMS    OF   NUMBERS. 


No. 

**  :i3: 

No.  !   Log. 

££'!  *«•    Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

1240 

093422 

1300  113943 

1360 

133539 

1420 

152288 

1 

093772  35 

1  114277 

33 

j 

133858 

32 

1 

152594 

30 

2 

094122   70  !    2.114611 

67 

2 

134177 

64 

2 

152900 

61 

3 

094471  105 

3  114944 

100    3  134496 

96 

3 

153205 

91 

4 

094820  i  140 

4  115278 

138 

4 

134814 

127 

4 

153510 

122 

6 

095169  !  175 

5 

115610 

167 

5 

135133 

159 

5 

153815 

152 

6 

095518  210 

6 

115943 

200  ' 

6 

135451 

191 

6 

154119 

183 

7 

095866 

245 

7 

116276 

23"3 

7 

135768 

223 

7 

154424 

213 

8 

096215 

280 

8 

116608 

267 

8 

136086 

255 

8 

154728 

244 

9 

096562 

315 

9 

116940 

300 

9 

136403 

287 

9 

155032 

274 

1250 

096910 

1310 

117271 

1370 

136721 

1430 

155336 

1 

097257 

35 

1  117603 

33!    1 

137037 

32 

1 

155640 

30 

2 

097604 

69 

2  117934 

66 

2 

187354 

63 

2 

155943 

60 

3 

097961 

104 

3  118265 

99 

3 

137670 

94 

3 

156246 

91 

4 

098297 

138 

4 

118595 

132! 

4 

137987 

126 

4 

156549 

121 

5 

098644 

173 

5 

118926 

165! 

5 

138303 

158 

5 

156852 

151 

6 

098990 

208 

6  j  119256 

198 

6 

138618 

189 

6 

157154 

181 

7 

099335 

242 

7 

119586 

231 

7 

138934 

221 

7 

157457 

211 

8 

099681 

277 

8 

119915 

264 

8 

139249 

252 

8 

157759 

242 

9 

100026 

311 

9 

120245 

297 

9 

139564 

284 

9 

158061 

272 

1260 

100370 

1820 

120574 

1380 

139879 

1440 

158362 

1 

100715 

34 

1 

120903 

88 

1 

140194 

31 

1 

158664 

30 

2 

101059 

69 

2 

121231 

66  i 

-   2 

140508 

63 

2 

158965 

60 

3 

101403 

103 

3 

121560 

98  i 

3 

140822 

94 

3 

159266 

90 

4 

101747 

187 

4 

121888 

131 

4 

141136 

125 

4 

159567 

120 

5 

102090 

172 

5 

122216 

164 

5 

141450 

157 

5 

159868 

150 

6 

102434 

206 

6 

122543 

197 

6 

141763 

188 

6 

160168 

180 

7 

102777 

240 

7 

122871 

230 

7 

142076 

219 

7 

160468 

210 

8 

103119 

275 

8 

123198 

262 

8 

142889 

251 

8 

160769 

240 

9 

103462 

309 

9 

123525 

295 

9 

142702 

282 

9 

161068 

270 

1270 

103804 

1330 

123852 

1390 

143015 

1450 

161368 

1 

104146 

34 

1 

124178 

33 

1 

143327 

31 

1 

161667 

30 

2 

104487 

68 

2 

124504 

65 

2 

143639 

62 

2 

161967 

60 

3 

104828 

102 

8 

124830 

98 

3 

143951 

93 

3 

162266 

89 

4 

105169 

136 

4 

125156 

130 

4 

144263 

125 

4 

162564 

119 

5 

105510 

170 

5 

125481 

163 

5 

144574 

156 

5 

162863 

149 

6 

105851 

204 

6 

125806 

195 

6 

144885 

187 

6 

163161 

179 

7 

106191 

238 

7 

126131 

228 

7 

145196 

218 

7 

163460 

209 

8 

106531 

272 

8 

126456 

260 

8 

145507 

249 

8 

163757 

239 

9 

106870 

306 

9 

126781 

293 

9 

145818 

280 

9 

164055 

269 

1280 

107210 

1340 

127105 

1400 

146128 

1460 

164358 

1 

107549 

34 

1 

127429 

32 

1 

146438 

31 

1 

164650 

30 

2 

107888 

67 

2 

127752 

65 

2 

146748 

62 

2 

164947 

59\ 

3 

108227 

101 

3 

128076 

97 

3 

147058 

93 

3 

165244 

89 

4 

108565 

135 

4 

128399 

129 

4 

147867 

124 

4 

165541 

119 

5 

108903 

169 

5 

128722 

161 

5 

147676 

155 

5 

165838 

148 

6 

109241 

203 

6 

129045 

194 

6 

147985 

186 

6 

166134 

178 

7 

109578 

237 

7 

129368 

226 

7 

148294 

217 

7 

166430 

207 

8 

109916 

270 

8 

129690 

258 

8 

148603 

248 

8 

166726 

237 

9 

110253 

304 

9 

130012 

291 

9 

148911 

279 

9 

167022 

2li7 

1290 

110590 

1350 

130334 

1410 

149219 

1470 

167317 

1 

110926 

34 

1 

130655 

32 

1 

149527 

31 

1 

167613 

29 

2 

111262 

67 

2 

130977 

64 

2 

149835 

61 

2 

167908 

59 

3 

111598 

101 

3  131298 

96 

3  150142 

92 

3 

168208 

88 

4 

111934 

134 

4  131619  128 

4  150449 

123    4 

168497 

118 

5 

112270 

168 

5 

131939  160 

5  150756 

154:'    5 

168792 

147 

6 

112605  201 

6 

132260  !  192 

6  151063 

184    6 

169086 

177 

7 

112940  235 

7  !  132580  !  224 

7  151370 

215    7 

169380 

206 

8 

113275 

268 

8  1329001256 

8  151676 

246    8 

169.674 

236 

9 

113609 

302 

9  133219  i  288 

9  151982 

277  ;!   9 

169968 

265 

LOGARITHMS   OF  NUMBERS. 


No, 

Log. 

SE 

No. 

Log. 

Prop 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop.] 
Part. 

1480 

170262 

1540 

187521 

1600 

204120 

1660 

220108 

1 

170555 

29 

1 

187803 

28 

1 

204391 

27 

1 

220370 

26 

2 

170848 

58 

2 

188084 

56 

2 

204662 

64 

2 

220631 

52 

3 

171141 

88 

3 

188366 

84 

3 

204933 

81 

3 

220892 

78 

4 

171434 

117 

4 

188647 

113 

4 

205204 

108 

4 

221153 

104 

5 

171726 

146 

5 

188928 

141 

5 

205475 

135 

5 

221414 

130 

6 

172019 

175 

6 

189209 

169 

6 

205745 

162 

6 

221675 

157 

7 

172311 

204 

7 

189490 

197 

7 

206016 

189 

7 

221936 

183 

8 

172608 

234 

8 

189771 

225 

8 

206286 

216 

8 

222196 

209 

9 

172895 

263 

9 

190051 

253 

9 

206556 

243 

9 

222456 

235 

1490 

173186 

1550 

190332 

1610 

206826 

1670 

222716 

1 

173478 

29 

1 

190612 

28 

1 

207095 

27 

1 

222976 

26 

2 

173769 

58 

2 

190892 

56 

2 

207365 

54 

2 

223236 

52 

3 

174060 

87 

3 

191171 

84 

3 

207634 

81 

3 

223496 

78 

4 

174351 

116 

4 

191451 

112 

4 

2079^)3 

108 

4 

223755 

104 

5 

174641 

145 

5 

191730 

140 

5 

208172 

135 

5 

224015 

130 

6 

174932 

175 

6 

192010 

168 

6 

208441 

162 

6 

224274 

156 

7 

175222 

204 

7 

192289 

196 

7 

208710 

188 

7 

224533 

182 

8 

175512 

233 

8 

192567 

224 

8 

208978 

215 

8 

224792 

208 

9 

175802 

261 

9 

192846 

252 

9 

209247 

241 

9 

225051 

234 

1500 

176091 

1560 

193125 

1620 

209515 

1680 

225309 

1 

176381 

29 

1 

193403 

28 

1 

209783 

27 

1 

225568 

26 

2 

176670 

58 

2 

193681 

56 

2 

210051 

54 

2 

225826 

52 

3 

176959 

86 

3 

193959 

83 

3 

210318 

80 

3 

226084 

77 

4 

177248 

115 

4 

194237 

111 

4 

210586 

107 

4 

226342 

103 

5 

177536 

144 

5 

194514 

139 

5 

210853 

134 

5 

226600 

129 

6 

177825 

173 

6 

194792 

166 

6 

211120 

161 

6 

226858 

155 

.7 

178113 

202 

7 

195069 

194 

7 

211388 

187 

7 

227115 

181 

8 

178401 

231 

8 

195346 

222 

8 

211654 

214 

8 

227372 

206 

9 

178689 

259 

9 

195623 

250 

9 

211921 

240 

9 

227630 

232 

1510 

178977 

1570 

195900 

1630 

212188 

1690 

227887 

1 

179264 

29 

1 

196176 

27 

1 

212454 

27 

1 

228144 

26 

2 

179552 

57 

2 

196452 

55 

2 

212720 

53 

2 

228400 

51 

3 

179839 

86 

3 

196729 

83 

3 

212986 

80 

3 

228657 

77 

4 

180126 

115 

4 

197005 

110 

4 

213252 

106 

4 

228913 

102 

5 

180413 

144 

5 

197281 

138 

5 

213518 

133 

5 

229170 

128 

6 

180699 

172 

6 

197556 

166 

6 

213783 

159 

6 

229426 

154 

7 

180986 

201 

7 

197832 

193 

7 

214049 

186 

7 

229682 

179 

8  181272 

230 

8 

198107 

221 

8 

214314 

212 

8 

229938 

205 

9  181558 

258 

9 

198382 

248 

9 

214579 

239 

9 

230193 

231 

1520  1  181844 

1580 

198657 

1640 

214844 

1700 

230449 

1  |  182129 

28    1 

198932 

27 

1 

215109 

26 

1 

230704 

25 

2 

182415 

571   2 

199206 

55 

2 

215373 

53 

2 

230960 

51  1 

3 

182700 

86 

3 

199481 

82 

3 

215638 

79 

3 

231215 

76 

4 

182985 

114 

4 

199755 

110 

4 

215902 

106 

4 

231470 

102 

5 

183270 

143 

5 

200029 

137 

5 

216166 

132 

5 

231724 

127 

6 

183554 

171 

6 

200303 

164  i 

6 

216430 

158 

6 

231979 

153 

7 

183839 

200 

7 

200577 

192 

7 

216694 

185 

7 

232233 

178 

8 

184123 

228 

8 

200850 

219 

8 

216957 

211 

8 

232488 

204 

9 

184407 

256 

9 

201124 

247 

9 

217221 

238 

9 

232742 

229 

1530 

184691 

1590 

201397 

1650 

217484 

1710 

232996 

1 

184975 

28 

1 

201670 

27 

1 

217747 

26 

1 

233250 

25 

2 

185259 

57 

2 

201943 

54 

2 

218010 

52 

2 

233504 

61 

3 

185542 

85 

3 

202216 

82 

3 

218273 

79 

3 

233757 

76 

4 

185825 

113 

4 

202488 

109 

4 

218535 

105 

4 

234011 

101 

5 

186108 

142 

5 

202761 

136 

5 

218798 

131 

5 

234264 

127 

6 

186391 

170 

6 

203033 

163 

6 

219060 

157 

6 

234517 

152 

7 

186674 

198 

7 

203305 

191 

7 

219322 

183 

7 

234770 

177 

8  !  186956 

227 

8 

203577 

218 

8 

219584 

210 

8 

235023 

202 

9  1  187239  255 

9 

203848 

245 

9 

219846 

236 

9 

235276 

228 

LOGARITHMS   OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Pare. 

No. 

Log. 

Prop. 
Part. 

1720 

235528 

1780 

250420 

1840 

264818 

1900 

278754 

1 

235781 

25 

1 

250664 

24 

1 

265054 

23 

1 

278982 

23 

2 

236033 

50 

2 

250908 

49 

2 

265290 

47 

2 

279210 

45 

3 

236285 

76 

3 

251151 

73 

3 

265525 

70 

3 

279439 

68 

4 

236537 

101 

4 

251395 

97 

4 

265761 

94 

4 

279667 

91 

5 

236789 

126 

5 

251638 

121 

5 

265996 

117 

5 

279895 

114 

6 

237041 

151 

6 

251881 

146 

6 

266232 

141 

6 

280123 

137 

7 

237292 

176 

7 

252125 

171 

7 

266467 

164 

7 

280351 

160 

8 

237544 

202 

8 

252367 

195 

8 

266702 

188 

8 

280578 

182 

9 

237795 

227 

9 

252610 

219 

9 

266937 

211 

9 

280806 

205  | 

1730 

238046 

1790 

252853 

1850 

267172 

1910 

281033 

1 

238297 

25 

1 

253096 

24 

1 

267406 

23 

1 

281261 

23 

2 

238548 

50 

2 

253338 

48 

2 

267641 

47 

2 

281488 

45 

3 

238799 

75 

3 

253580 

73 

3 

267875 

70 

3 

281715 

68 

4 

239049 

100 

4 

^53822 

97 

4 

268110 

94 

4 

281942 

91 

5 

239299 

125 

5 

254064 

121 

5 

268344 

117 

5 

282169 

113 

6 

239550 

150 

6 

254306 

145 

6 

268578 

141 

6 

282395 

136 

7 

239800 

175 

7 

254548 

170 

7 

268812 

164 

7' 

282622 

159 

8 

240050 

200 

8 

254790 

194 

8 

269046 

188 

3 

282849 

181 

9 

240300 

225 

9 

255031 

218 

9 

269279 

211 

9 

283075 

204 

1740 

240549 

1800 

255273 

1860 

269513 

1920 

283301 

1 

240799 

25 

1 

255514 

24 

1 

269746 

23 

1 

283527 

23 

2 

241048 

50 

2 

255755 

48 

2 

269980 

47 

2 

283753 

45 

3 

241297 

75 

3 

255996 

72 

3 

270213 

70 

3 

283979 

68 

4 

241546 

100 

4 

256236 

96 

4 

270446 

93 

4 

284205 

90 

5 

241795 

124 

5 

256477 

120 

5 

270679 

116 

5 

284431 

113 

6 

242044 

149 

6 

256718 

144 

6 

270912 

140 

6 

284656 

135 

7 

242293 

174 

7 

256958 

168 

7 

271144 

163 

7 

284882 

•158 

8 

242541 

199 

8 

257198 

192 

8 

271377 

186 

8 

285107 

180 

9 

242790 

223 

9 

257489 

216 

9 

271609 

210 

9 

285332 

203 

1750 

243038 

1810 

257679 

1870 

271842 

1930 

285557 

1 

243286 

25 

1 

257918 

24 

1 

272074 

23 

1 

285782 

22 

2 

243534 

50 

2 

258158 

48 

2 

272306 

46 

2 

286007 

45 

3 

243782 

74 

3 

258398 

72 

3 

272538 

70 

3 

286232 

67 

4 

244030 

99 

4 

258637 

96 

4 

272776 

93 

4 

286456 

89 

5 

244277 

124 

5 

258877 

120 

5 

273001 

116 

5 

286681 

112 

6 

244524 

149 

6 

259116 

144 

6 

273233 

139 

6 

286905 

134 

7 

244772 

174 

7 

259355 

167 

7 

273464 

162 

7 

287130 

157 

8 

245019 

198 

8 

259594 

192 

8 

273696 

186 

8 

287354 

179 

9 

245266 

222 

9 

259833 

215 

9 

273927 

209 

9 

287578 

202 

1760 

245513 

1820 

260071 

1880 

274158 

1940 

287802 

1 

245759 

25 

1 

260310 

24 

1 

274389 

23 

1 

288025 

22 

2 

246006 

49 

2 

260548 

48 

2 

274620 

46 

2 

288249 

45 

3 

246252 

74 

3 

260787 

71 

3 

274850 

69 

3 

288473 

67 

4 

246499 

98 

4 

261025 

95 

4 

275081 

92 

4 

288696 

89 

5 

246745 

123 

5 

261263 

119 

5 

275311 

115 

5 

288920 

112 

6 

246991 

148 

6 

261501 

143 

6 

275542 

138 

6 

28*9143 

134 

7 

247236 

173 

7 

261738 

167 

7 

275772 

161 

7 

289366 

156 

8 

247482 

197 

8 

261976 

191 

8 

276002 

184 

8 

289589 

178 

9 

247728 

221 

9 

262214 

214 

9 

276232 

207 

9 

289812 

201 

1770 

247973 

1830 

262451 

1890 

276462 

1950 

290035 

1 

248219 

25 

1 

262688 

24 

1 

276691 

23 

1 

290257 

22 

2 

248461 

49 

2 

262925 

47 

2 

276921 

46 

2 

290480 

44 

3 

248709 

74 

3 

263162 

71 

3 

277151 

69 

3 

290702 

67 

4 

248954 

98 

4 

263399 

95 

4 

277380 

92 

4 

290925 

89 

5 

249198 

123 

5 

263636 

118 

5 

277609 

115 

5 

291147 

111 

6 

249443 

147 

6 

263873 

142 

6 

277838 

138 

6 

291369 

133 

7 

249687 

172 

7 

264109 

166 

7 

278067 

161 

7 

291591 

156 

8 

249932 

196 

8 

264345 

190 

8 

278296 

183 

8 

291813 

178 

9 

250176 

220 

9 

264582 

213 

9 

278525 

206 

9 

292034 

200 

LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

1960 

292256 

2020 

305351 

2080 

318063 

2140 

330414 

1 

292478 

22 

.1 

305566 

21 

1 

318272 

21 

1 

330617 

20 

2 

292699 

44 

2 

305781 

43 

2 

318481 

42 

2 

330819 

40 

3 

292920 

66 

3 

305996 

64 

3 

318689 

63 

3 

331022 

61 

4 

293141 

88 

4 

306211 

86 

4 

318898 

83 

4 

331225 

81 

5 

293363 

110 

5 

306425 

107 

5 

319106 

104 

5 

331427 

101 

6 

293583 

133 

6 

306639 

129 

6 

319314 

125 

6 

331630 

121 

7 

293804 

155 

7 

306854 

150 

7 

319522 

146 

7 

331832 

141 

8 

294025 

177 

8 

307068 

172 

8 

319730 

167 

8 

332034 

162 

9 

294246 

199 

9 

307282 

193 

9 

319938 

188 

9 

332236 

182 

1970 

294466 

2030 

307496 

2090 

320146 

2150 

332438 

I 

1 

294687 

22 

1 

307710 

21 

1 

320354 

.  21 

1 

332640 

20 

2 

294907 

44 

2 

307924 

43 

2  320562 

41 

2 

332842 

40 

3 

295127 

66 

3 

308137 

64 

3 

320769 

62 

3 

333044 

60 

4 

295347 

88 

4 

308351 

85 

4 

320977 

83 

4 

333246 

81 

5 

295567 

110 

5 

308564 

107 

5 

321184 

104 

5 

333447 

101 

6 

295787 

132 

6 

308778 

128 

6 

321391 

125 

6 

333649 

121 

7 

296007 

154 

7 

308991 

149 

7 

321598 

145 

7 

333850 

141 

8 

296226 

176 

8 

309204 

171 

8 

321805 

166 

8 

334051 

161 

9 

296446 

198 

9 

309417 

192 

9 

322012 

187 

9 

334253 

181 

1980 

296665 

2040 

309630 

2100 

322219 

2160 

334454 

1 

296884 

22 

1 

309843 

21 

1 

322426 

21 

1 

334655 

20 

2 

297104 

44 

o 

dfi 

310056 

43 

2 

322633 

41 

2 

334856 

40 

3 

297323 

66 

3 

310268 

64 

3 

322839 

62 

3 

335056 

60 

4 

297542 

88 

4 

310481 

85 

4 

323046 

82 

4 

335257 

80 

5 

297761 

109 

5 

310693 

106 

5 

323252 

103 

5 

335458 

100 

(5 

297979 

131 

6 

310906 

127 

6 

323458 

124 

6 

335658 

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1 

298198 

153 

7 

311118 

148    7 

323665  144 

7 

335859 

140 

8 

298416 

175 

8 

311330 

170 

8 

323871  165 

8 

336059 

160 

9 

298635 

197 

9 

311542 

191 

9 

324077 

186 

9 

336260 

180 

1990 

298853 

2050 

311754 

2110 

324282 

2170 

336460 

1 

299071 

22 

1 

311966 

21 

1 

324488 

21 

1 

336660 

20 

2 

299289 

44 

2 

312177 

42 

2 

324694 

41 

2 

336860 

40 

3 

299507 

65 

3 

312389 

63 

3 

324899 

62 

3 

337060 

60 

4 

299725 

87 

4 

312600 

84 

4 

325105 

82 

4 

337260 

80 

5 

299943 

109 

5 

312812 

106 

5 

325310 

103 

5 

337459 

100 

6 

300160 

131 

6 

313023 

127 

6 

325516 

123 

6 

337659 

120 

7 

300378 

153 

7 

313234 

148 

7 

325721 

144 

7 

337858 

140 

8 

300595 

174 

8 

313445 

160 

8 

325926 

164 

8 

338058 

160 

9 

300813 

196 

9 

313656 

190 

9 

326131 

185 

9 

338257 

180 

2000 

301030 

2060 

313867 

2120 

326336 

2180 

338456 

1 

301247 

22 

1 

314078 

21 

1 

326541 

20 

1 

338656 

20 

2 

301464 

43 

2 

314289 

42 

2 

326745 

41 

2 

338855 

40 

3 

301681 

65 

3 

314499 

63 

3 

326950 

61 

3 

339054 

60 

4 

301898 

87 

4 

314710 

84 

4 

327155 

82 

4 

339253 

80 

5 

302114 

108 

5 

314920 

105 

5 

327359 

102 

5 

339451 

100 

6 

302331 

130 

6 

315130 

126 

6 

327563 

123 

6 

339650 

119 

7 

302547 

152 

7 

315340 

147 

7 

327767 

143 

7 

339849 

139 

8 

302764 

173 

8 

315550 

168 

8 

327972 

164 

8 

340047 

159 

9 

302980 

195 

9 

315760 

189 

9 

328176 

184 

9 

340246 

179 

2010 

303196 

2070 

315970 

2130 

328380 

2190 

340444 

1 

303412 

22 

1 

316180 

21 

1 

328583 

20 

1 

340642 

20 

•  2 

303628 

43 

2 

316390 

42 

2 

328787 

41 

2 

340841 

40 

3 

303844 

65 

3 

316599 

63 

3 

328991 

61 

3 

341039 

59 

4 

304059 

86 

4 

316809 

84 

4 

329194 

81 

4 

341237 

79 

5 

304275 

108 

6 

317018 

105 

5 

329398 

102 

5 

341435 

99 

6 

304490 

129 

6 

317227 

126 

6 

329601 

122 

6 

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119 

7 

304706 

151 

7 

317436 

147 

7 

329805 

142 

7 

341830 

139 

8 

304921 

172 

8  :  317645 

168 

8 

330008 

163 

8 

342028 

158 

9 

305136 

194 

9  317854 

189 

9 

330211 

183  :   9 

342225 

178 

LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

s& 

No. 

Log. 

g£  N°- 

Log. 

Prop 
Pari 

No. 

Log. 

Prop. 
Part. 

2200 

342423 

2260 

354108 

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365488 

2380 

376577 

1 

342620 

20 

1 

354301 

19 

1 

365675 

19 

1 

376759 

18 

2 

342817 

39 

2 

354493 

38 

2 

365862 

37 

2 

376942 

36 

3 

343014 

59 

3 

354685 

58 

c 

t. 

366049 

56 

3 

377124 

55 

4 

343212 

79 

4 

354876 

77 

1 

366236 

75 

4 

377306 

73 

5 

343409 

99 

• 

355068 

96 

f 

366423 

93 

5 

377488 

91 

6 

343606 

118 

I 

355260 

115 

e 

366610 

112 

6 

377670 

109 

7 

343802 

138 

7 

355452 

134 

7 

366796 

131 

7 

377852 

127 

8 

343999 

158 

£ 

355643 

154 

8 

366983 

150 

8 

378034 

146 

9 

344196 

178 

9 

355834 

173 

c 

367169 

168 

9 

378216 

164 

2210 

344392 

i2270 

356026 

2330 

367356 

2390 

378398 

1 

344589 

20 

1 

356217 

19 

1 

367542 

19 

1 

378580 

18 

2 

344785 

39 

2 

356408 

38 

2 

367729 

37 

2 

378761 

36 

3 

344981 

59 

3 

356599 

57 

3 

367915 

56 

3 

378943 

55 

4 

345178 

78 

4 

356790 

76 

4 

368101 

75 

4 

379124 

73 

5 

345374 

98 

5 

356981 

95 

B 
t> 

368287 

93 

5 

379306 

91 

6 

345570 

118 

6 

357172 

115 

6 

368473 

112 

6 

379487 

109 

7 

345766 

137 

7 

357363 

134 

7 

368659 

130 

7 

379668 

127 

8 

345962 

157 

8 

357554 

153 

8 

368844 

149 

8 

379849 

146 

9 

346157 

176 

9 

357744 

172 

9 

369030 

167 

9 

380030 

164 

2220 

346353 

2280 

357935 

2340 

369216 

2400 

380211 

1 

346549 

19    1 

358125 

19 

1 

369401 

19 

1 

380392 

18 

2 

346744 

39    2 

358316 

38 

2 

369587 

37 

2 

380573 

36 

3 

346939 

58    3 

358506 

57 

3 

369772 

56 

3 

380754 

55 

4 

347135 

78 

4 

358696 

76 

4 

369958 

74 

4 

380934 

73 

5 

347330 

97 

5 

358886 

95 

5 

370143 

93 

5 

381115 

91 

6 

347525 

117 

6 

359076 

114 

6 

370328 

111 

6 

381296 

109 

"7 

347720 

137 

7 

359266 

133 

7 

370513 

130 

7 

381476 

127 

8 

347915 

156 

8 

359456 

152 

8 

370698 

148 

8 

381656 

145 

9 

348110 

175 

9 

359646 

171 

9 

370883 

167 

9 

381837 

163 

2230 

848305 

2290 

359835 

2350 

371068 

2410  '38201  7 

1 

848500 

19 

1 

360025 

19 

1 

371253 

18 

1  382197 

18 

2 

848694  3d 

2 

360215 

38 

2 

371437 

37 

2  382377 

36 

3 

348889 

58 

3 

360404 

57 

3 

371622 

55 

3  382557 

54 

4 

349083 

78 

4 

360593 

76 

4 

371806 

74 

4 

382737 

72 

5 

349278 

97 

5 

360783 

95 

5 

371991 

92 

5 

382917 

90 

6 

349472 

117 

6 

360972 

114 

6 

372175 

111 

6 

383097 

108 

7 

349666 

137 

7 

361161 

133 

7 

372360 

129 

7 

383277 

126 

8 

349860 

156 

8 

361350 

152 

8 

372544 

148 

8 

383456 

144 

9 

350054 

175 

9 

361539 

171 

9 

372728 

166 

9 

383636 

162 

2240 

350248 

2300 

361728 

2360 

372912 

2420 

383815 

1 

350442 

19 

1 

361917 

19 

1 

373096 

18 

1 

383995 

18 

2 

350636 

39 

2 

362105 

38 

2 

373280 

37 

o 

384174 

36 

3 

350829 

58 

3 

362294 

56 

3 

373464 

55 

3 

384353 

54 

4 

351023 

77 

4 

362482 

75 

4 

373647 

74 

4 

384533 

72 

5 

351216 

97 

5 

362671 

94 

5 

373831 

92 

5 

384712 

90 

6  351410 

116 

6 

362859 

113 

6 

374015 

110 

6 

384891 

108 

7  351603 

135 

7 

363048 

132 

7 

374198 

129 

7 

385070 

126 

8  351796 

155 

8 

363236 

151 

8 

374382 

147 

8 

385249 

144 

9  351989 

174 

9 

363424 

170 

9 

374565 

166 

9 

385428 

162 

2250  352182 

2310 

363612 

2370 

374748 

430 

385606 

1  352375 

19 

1 

363800 

19 

1 

374932 

18 

1 

385785 

18 

2  i  352568 

38 

2 

363988 

37 

2 

375115 

37 

2 

385964 

35 

3  352761 

58 

3 

364176 

56 

3 

375298 

55 

3 

386142 

53 

4  |  352954 

77 

4 

364363 

75 

4 

375481 

73 

4 

386321 

71 

5  353147 

96 

5 

364551 

94 

5 

375664 

92 

5 

386499 

89 

6  353339 

115 

6 

364739 

112 

6 

375846 

10 

6 

386677 

07 

7  353532 

134 

7 

364926 

131 

7 

376029 

28 

7 

386856 

25 

8  353724 

154 

8 

365113 

150 

8 

376212 

47 

8 

387034 

43 

9  353916 

173 

9 

365301 

169 

9 

376394 

65 

'9 

387212 

61 

LOGARITHMS   OF   NUMBERS. 


No. 

Log. 

sa 

No.     Log. 

Prop. 
Part. 

No. 

Log. 

?rop. 
Part. 

No. 

Log. 

Prop. 
Part. 

2440 

387390 

2500  397940 

2560 

408240 

2620 

418301 

1 

387568 

18 

1  398114 

17 

1 

408410 

17 

1 

418467 

17 

2 

387746 

36 

2  398287 

35 

2 

408579 

34 

2 

418633 

33 

3 

387923 

53 

3 

398461 

53 

3 

408749 

51 

3 

418798 

50 

4 

388101 

71 

4 

398634 

69 

4 

408918 

68 

4 

418964 

66 

5 

388279 

89 

5 

398808 

87 

5 

409087 

85 

5 

419129 

83 

6 

388456 

107 

6 

398981 

104 

6 

409257 

102 

6 

419295 

99 

7 

388634 

125 

7 

399154 

121 

7 

409426 

119 

7 

419460 

116 

8 

388811 

142 

8 

399327 

138 

8 

409595 

136 

8 

419625 

132 

9 

388989 

160 

9 

399501 

156 

9 

409764 

153 

9 

419791 

149 

2450 

389166 

2510 

399674 

2570 

409933 

2630 

419956 

1 

389343 

18 

1 

399847 

17 

1 

410102 

17 

1 

420121 

16 

2 

389520 

36 

2 

400020 

35 

2 

410271 

34 

2 

420286 

33 

3 

389697 

53 

3 

400192 

53 

3 

410440 

50 

3 

420451 

49 

4 

389875 

71 

4 

400365 

69 

4 

410608 

67 

4 

420616 

66 

5 

390051 

89 

5 

400538 

87 

5 

410777 

84 

5 

420781 

82 

6 

390228 

107 

6 

400711 

104 

6 

410946 

101 

6 

420945 

99 

7 

390405 

125 

7 

400883 

121 

7 

411114 

118 

7 

421110 

115 

8 

390582 

142 

8 

401056 

138 

8 

411283 

135 

8 

421275 

132 

9 

390759 

160 

9 

401228 

156 

9 

411451 

152 

9 

421439 

148 

2460 

390935 

2520 

401400 

2580 

411620 

2640 

421604 

1 

391112 

18 

1 

401*573 

17 

1 

411788 

17 

1 

421768 

16 

2 

391288 

35 

2 

401745 

34 

2 

411956 

34 

2 

421933 

33 

3 

391464 

53 

3 

401917 

52 

3 

412124 

50 

3 

422097 

49 

4 

391641 

70 

4 

402089 

69 

4 

412292 

67 

4 

422261 

66 

5 

391817 

88 

5 

402261  j  86 

5 

412460 

84 

5 

422426 

82 

6 

391993 

106 

.   6 

402433  |  103 

6 

412628 

101 

6 

422590 

99 

7 

392169 

123 

7 

402605 

120 

7 

412796 

118 

7 

422754 

115 

8 

392345 

141 

8 

402777 

138 

8 

412964 

135 

8 

422918 

132 

9 

392521 

158 

9 

402949 

155 

9 

413132 

152 

9 

423082 

148 

2470 

392G97 

2530 

403120 

2590 

413300 

2650 

423246 

1 

392873 

18 

1 

403292 

17 

1 

413467 

17 

1 

423410 

16 

2 

393048 

35 

2 

403464 

34 

2 

413635 

33 

2 

423573 

33 

3 

393224 

53 

3 

403635 

52 

8 

413802 

50 

3 

423737 

49 

4 

393400 

70 

4 

403807 

69 

4 

413970 

67 

4 

423001 

65 

5 

393575 

88 

5 

403978 

86 

5 

414137 

84 

5 

424064 

81 

6 

393751 

106 

6 

404149 

103 

6 

414305 

101 

6 

424228 

98 

7 

393926 

123 

7 

404320 

120 

7 

414472 

117 

7 

424392 

114 

8 

394101 

141 

8 

404492 

137 

8 

414639 

134 

8 

424555 

131 

9 

394276 

158 

9 

404663 

154 

9 

414806 

151 

9 

424718 

147 

2480 

394452 

2540 

404834 

2600 

414973 

2660 

424882 

1 

394627 

17 

1 

405005 

17 

1 

415140 

17 

1 

425045 

16 

2 

394802 

35 

2 

405175 

34 

2 

415307 

33 

2 

425208 

33 

3 

394977 

53 

3 

405346 

51 

3 

415474 

50 

3 

425371 

49 

4 

395152 

70 

4 

405517 

68 

4 

415641 

67 

4 

425534 

65 

5 

395326 

87 

5 

405688 

85 

5 

415808 

84 

5 

425697 

81 

6 

395501 

104 

6 

405858 

102 

6 

415974 

101 

6 

425860 

98 

7 

395676 

122 

7 

406029 

119 

7 

416141 

117 

7 

426023 

114 

8 

395850 

139 

.  8 

406199 

136 

8 

416308 

134 

8 

426186 

130 

9 

396025 

157 

9 

406370 

153 

9 

416474 

150 

9 

426349 

147 

2490 

396199 

2550 

406540 

2610 

416640 

2670 

426511 

1 

396374 

17 

1 

406710 

17 

1 

416807 

17 

1 

426674 

16 

2 

396548 

35 

2 

406881 

34 

2 

416973 

33 

2 

426836 

33 

3 

396722 

53 

3 

407051 

51 

3 

417139 

50 

3 

426999 

49 

4 

396896 

70 

4 

407221 

68 

4 

417306 

66 

4 

427161 

65 

5 

397070 

87 

5 

407391 

85 

5 

417472 

83 

5 

427324 

81 

6 

397245 

104 

6 

407561 

102 

6 

417638 

100 

6 

427486 

98 

7 

397418 

122 

7  I  407731 

119 

7 

417804 

116 

7 

427648 

114 

8 

397592 

139 

8  407900  136 

8 

417970 

133 

8 

427811 

130 

9 

397766 

157 

9  408Q70  i  153 

9 

418135 

149 

9 

427973 

147 

10 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

,0, 

Prop. 

Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Frop, 

Part. 

2680 

428135 

2740 

437751 

2800 

447158 

2860 

456366 

1 

428297 

16 

1 

437909 

16 

1 

447313 

15 

1 

456518 

15 

2 

428459 

32 

2 

438067 

32 

2 

447468 

31 

2 

456670 

30 

3 

4li8621 

4« 

3 

438226 

47 

3 

447623 

46 

3 

456821 

46 

4 

428782 

Go 

4 

438384 

63 

4 

447778 

62 

4 

456973 

61 

5 

428944 

81 

5 

438542 

79 

5 

447933 

77 

5 

457125 

76 

6 

429106 

97 

6 

438700 

95 

6 

448088 

93 

6 

457276 

91 

7 

429268 

113 

7 

438859 

111 

7 

448242 

108 

7 

457428 

106 

8 

429429 

129 

8 

439017 

127 

8 

448397 

124 

8 

457579 

122 

9 

429591 

145 

9 

439175 

143 

9 

448552 

139 

9 

457730 

137 

2690 

429752 

2750 

439338 

2810 

448706 

2870 

457882 

1 

429914 

16 

1 

439491 

16 

1 

448861 

15 

I 

458033,  15 

2 

430075 

32 

2 

439648 

32 

2 

449015 

31 

o 

458184J  30 

3 

430236 

48 

3 

439806 

47 

3 

449170 

46 

3 

458336 

45 

4 

430398 

65 

4 

439964 

63 

4 

449324 

62 

A 

458487 

61 

5 

430559 

81 

5 

440122 

79 

5 

449478 

77 

5 

458638 

76 

6 

430720 

97 

6 

440279 

95 

6 

449633 

92 

6 

458789 

91 

7 

430881 

113 

7 

440437 

111 

7 

449787 

108 

7 

458940 

106 

8 

431042 

129 

8 

440594 

126 

8 

449941 

123 

8 

459091 

121 

9 

431203 

145 

9 

440752 

142 

9 

450095 

139 

9 

459242 

136 

2700 

431364 

|  2760 

440909 

2820 

450249 

2880 

459392 

1 

431525 

16    1 

441066 

16 

1 

450403 

15 

1 

459543 

15 

2 

431685 

32    2 

441224 

31 

2 

450557 

31 

2 

459694 

30 

3 

431846 

48 

3 

441381 

47 

3 

450711 

46 

3 

459845 

45 

4 

432007 

64 

4 

441538 

63 

4  450865 

62 

4 

459995 

61 

5 

432167 

80 

5 

441695 

78 

5 

451018 

77 

5 

460146 

76 

6 

432328 

96 

6 

441852 

94 

6 

451172 

92 

6 

460296 

91 

7 

432488 

112 

7 

442009 

110 

7 

451326 

108 

7 

460447  106 

8 

432649 

128 

8 

442166 

126 

8 

451479 

123 

8 

460597 

121 

9 

432809 

144 

9 

442323 

141 

9 

451633 

139 

9 

460747 

136 

2710 

432969 

2770 

442480 

2830 

451786 

2890 

460898 

1 

433129 

16 

1 

442636 

16 

1 

451940 

15 

1 

461048 

15 

2 

433290 

32 

2 

442793 

31 

2 

452093 

31 

2 

461198 

30 

3 

433450 

48 

3 

442950 

47 

3 

452247 

46 

3 

461348 

45 

4 

433610 

64 

4 

443106 

63 

4 

452400 

61 

4 

461498 

60 

5 

433770 

80 

5 

443263 

78 

5 

452553 

77 

5 

461649 

75 

6 

433930 

96 

6 

443419 

94 

6 

452706 

92 

6 

461799 

90 

7 

434090 

112 

7 

443576 

110 

7 

452859 

107 

7 

461948 

105 

8 

434249 

128 

8 

443732 

126 

8 

453012 

123 

8 

462098 

120 

9 

434409 

144 

9 

443888 

141 

9 

453165 

138 

9 

462248 

135 

2720 

434569 

2780 

444045 

2840 

453318 

2900 

462398 

1 

434728 

16 

1 

444201 

16 

1 

453471 

15 

1 

462548 

15 

2 

434888 

32 

2 

444357 

31 

2 

453624 

31 

2 

462697 

30 

3 

435048 

48 

3 

444513 

47 

-  3 

453777 

46 

3 

462847 

45 

4 

435207 

64 

4 

444669 

62 

4 

453930 

61 

4 

462997 

60 

5 

435366 

80 

5 

444825 

78 

5 

454082 

77 

5 

463146 

75 

6 

435526 

96 

6 

444981 

94 

6 

454235 

92 

6 

463296 

90 

7 

435685 

112 

7 

445137 

109 

7 

454387 

107 

7 

463445 

105 

8 

435844 

128 

8 

445293 

125 

8 

454540 

123 

8 

463594 

120 

9 

436003 

144 

9 

445448 

140 

9 

454692 

138 

9 

463744 

135 

2730 

436163 

2790 

445604 

2850 

454845 

2910 

463893 

1 

436322 

16 

1 

445760 

16 

1 

454997 

15 

1 

464042 

15 

2 

436481 

32 

2 

445915 

31 

2 

455149 

30 

2 

464191 

30 

3 

436640 

47 

3 

446071 

47 

3 

455302 

46 

3  I  464340 

45 

4 

436798 

63 

4 

44G226 

62 

4 

455454 

61 

4 

4G4489 

60 

5 

436957 

79 

5 

446382 

78 

5 

455606 

76 

5 

464639 

75 

6 

437116 

95 

6 

446537 

94 

6 

455758 

91 

6 

464787 

90 

7 

437275 

111 

7 

446692 

109 

7 

455910 

106 

7 

464936 

105 

8 

437433 

127 

8 

446848 

125 

8 

456062 

122 

8 

465085 

120 

9 

437592 

143 

9 

447003 

140 

9 

456214 

137 

9 

465234 

135 

LOGARITHMS    OF   NUMBERS. 


11 


No. 

Log. 

Prop. 
Part. 

No. 

LOS-   IS 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

2U20 

465383 

2980 

474216 

3040 

482874 

3100 

491362 

1 

465532|  15 

1 

474362 

15 

1 

4S3016 

14 

1 

491502 

14 

2 

46568U 

30 

2 

474508 

29 

2 

483159 

28 

2 

491642 

28 

3 

465829 

44 

3 

474653 

44 

3 

483302 

43 

3 

491782 

42 

4 

465977 

59 

4 

474799 

58 

4 

483445 

57 

4 

491922 

56 

5 

466126 

74 

5 

474944 

73 

5 

483587 

71 

5 

492062 

70. 

6 

466274 

89 

6 

475090 

88 

6 

483730 

85 

6 

492201 

84 

7 

466423 

104 

7 

475235 

102 

7 

483872 

99 

7 

492341 

98 

8 

466571 

118 

8 

475381 

117 

8 

484015 

114 

8 

492481 

112 

9 

466719 

133 

9 

475526 

131 

9 

484157 

128 

9 

492621 

126 

2930 

466868 

2990 

475671 

3050 

484300 

3110 

492760 

1 

467016 

15 

1 

475816 

15 

,  1 

484442 

14 

1 

492900 

14 

2 

467164 

30 

2 

475962 

29 

2 

484584 

28 

2 

493040 

28 

3 

467312 

44 

3 

476107 

43 

3 

484727 

43 

3 

493179 

42 

4 

467460 

59 

4 

476252 

58 

4 

484869 

57 

4 

493319 

56 

5 

467608 

74 

5 

476397 

72 

5 

485011 

71 

5 

493458 

70 

6 

467756 

89 

6 

47G542 

87 

6 

485153 

85 

6 

493597 

84 

7 

467904 

104 

7 

476687 

101 

7 

485295 

99 

7 

493737 

98 

8 

468U52 

118 

8 

476832 

116     8 

485437 

114 

8 

493876 

142 

9 

468200 

133 

9 

476976 

130  II   9 

485579 

128 

9 

494015 

126 

2940 

468347 

3000 

477121 

3060 

485721 

3120 

494155 

1 

468495 

15 

1 

477266 

14 

1 

485863 

14 

1 

494294 

14 

2 

468643 

30 

2 

477411 

29 

2 

486005 

28 

2 

494433 

28 

3  468790 

44 

3 

477555 

43 

3 

486147 

43 

3 

494572 

41 

\ 

868938 
469085 

59 
74 

4 
5 

477700 
477844 

58 

7%>  i 

4 

5 

486289 
486430 

57 
71 

4 
5 

494711 
494850 

56 
69 

6 

469233 

89 

6 

477989 

87  i 

6 

486572 

85 

6 

494989 

83 

7 

469380 

104 

7 

478133 

101  !    7 

486714 

99 

7 

495128 

97 

8 

469527 

118 

8 

478278 

116 

8 

486855 

114 

8 

495267 

111 

9 

469675 

133 

9 

478422 

130 

9 

486997 

128 

9 

•495406 

125 

2950 

469822 

3010 

478566 

3070 

487138 

3130 

495544 

i 

469969 

15 

1 

478711 

14 

1 

487280 

14 

1 

495683 

14 

2 

470116 

29 

2 

478855 

29 

2 

487421 

28 

2 

495822 

28 

3  470263 

44 

3 

478999 

43 

3 

487563 

42 

3 

495960 

41 

4  470410 

59 

4 

479143 

68 

4 

487704 

57 

4 

496099 

56 

5  1  470557 

74 

5 

479287 

72 

5 

487845 

71 

5 

496237 

69 

6  1  470704 

88 

6 

479431 

86 

6 

487986 

85 

6 

496376 

83 

7 

470851 

103 

7 

479575 

101 

7 

488127 

99 

7 

496514 

97 

8 

470998 

118 

8 

479719 

115  'I   8 

488269 

113 

8 

496653 

111 

9 

471145 

132 

9 

479863 

130    9 

488410 

127 

9 

496791 

125 

2960 

471292 

3020 

480007 

3080 

488551 

3140 

496930 

1 

471438 

15 

1 

480151 

14 

1 

488692 

14 

1 

497068 

14 

2 

471585 

29 

2 

480294 

29 

2 

488833 

28 

2 

497206 

28 

3 

471732 

44 

3 

480438 

43 

3 

488973 

42 

3 

497344 

41 

4 

471878 

59 

4  !  480582 

58 

4 

489114 

56 

4 

497482 

55 

5 

472025 

73 

5 

480725 

72 

5 

489255 

70 

5 

497621 

69 

6 

472171 

88 

6 

480869 

86 

6 

489396 

84 

6 

497759 

83 

7 

472317 

102 

7 

481012 

101 

7 

489537 

98 

7 

497897 

97 

8 

472464 

117 

8 

481156 

115 

8 

489677 

112 

8 

498035 

110 

9 

472610 

132 

9 

481299 

130 

9 

489818 

126 

9 

498173 

124 

2970 

472756 

3030 

481443 

3090 

489958 

3150 

498311 

1 

472903 

15 

1 

481586 

14 

1 

490099 

14 

1 

498448 

14 

2  473049 

29 

2 

481729 

29 

2 

490239 

28 

2 

498586 

28 

3  !  473195 

44 

3 

481872 

43 

3 

490380 

42 

3 

498724 

41 

4  ,  473341 

59 

4 

482016 

57 

4 

490520 

56 

4 

498862 

55 

5  i  473487 

73 

5 

482159 

71 

5 

490661 

70 

5 

498999 

69 

6  473633 

88 

6 

482302 

86 

6 

490801 

84 

6 

499137 

83 

7  473779 

102 

7 

482445 

100 

7 

490941 

98 

7 

499275 

97 

8  473925 

117 

8 

482588 

114 

8 

491081 

112 

8 

499412 

110 

9  474070  132 

9 

482731 

129 

9  491222 

126 

9 

499550 

124 

-  --—    '    ii 

,1                    j 

i 

12 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop.! 
Part. 

3160 

499687 

3220 

507856 

3280 

515874 

3340 

523746 

1 

499824 

14 

1 

507991 

13 

1 

516006 

13 

1 

523876 

13 

2 

499962 

27 

2 

508125 

27 

2 

516139 

26 

2 

524006 

26 

3 

500099 

41 

3 

508260 

40 

3 

516271 

40 

3 

524136 

39 

4 

500236 

55 

4 

508395 

54 

4 

516403 

53 

4 

524266 

52 

5 

500374 

68 

5 

508530 

67 

5 

516535 

66 

5 

524396 

65 

6 

500511 

82 

6 

508664 

81 

6 

516668 

79 

6 

524526 

78 

7 

500648 

96 

7 

508799 

94 

7 

516800 

92 

7 

524656 

91 

8 

500785 

110 

8 

508933 

108 

8 

516932 

106 

8 

524785 

104 

9 

500922 

123 

9 

509068 

121 

9 

517064 

119 

9 

524915 

117 

3170 

501059 

3230 

509202 

3290 

517196 

3350 

525045 

1 

501196 

14 

1 

509337 

13 

1 

517328 

13 

1 

525174 

13 

2 

501333 

27 

2 

509471 

27 

2 

517460 

26 

2 

525304 

26 

3 

501470 

41 

3 

509606 

40 

3 

517592 

40 

3 

525434 

39 

4 

501607 

55 

4 

509740 

54 

4 

517724 

53 

4 

525563 

52 

5 

501744 

68 

5 

509874 

67 

5 

517855 

66 

5 

525693 

65 

6 

501880 

82 

6 

510008 

81 

6 

517987 

79 

6 

525822 

78 

7 

502017 

96 

7 

510143 

94 

7 

518119 

92 

7 

525951 

91 

8 

502154 

110 

8 

510277 

108 

8 

518251 

106 

8 

526081 

104 

9 

502290 

123 

9 

510411 

121 

0 

518382 

119 

9 

526210 

117 

3180 

502427 

3240 

510545 

3300 

518514 

3360 

526339 

1 

502564 

14 

1 

510679 

13 

1 

518645 

13 

1 

526468 

13 

2 

502700 

27 

2 

510813 

27 

2 

518777 

26 

2 

526598 

26 

3 

502837 

41 

3 

510947 

40 

3 

518909 

39 

3 

526727 

39 

4 

502973 

54 

4 

511081 

54 

4 

519040 

52 

4 

526856 

52 

6 

503109 

68 

5 

511215 

67 

5 

519171 

66 

5 

526985 

65 

6 

503246 

82 

6 

511348 

80 

6 

519303 

79 

6 

527114 

78 

7 

503382 

95 

7 

511482 

94 

7 

519434 

92 

7 

527243 

91 

8 

503518 

109 

8 

511616 

107 

8 

519565 

105 

8 

527372 

104 

9 

503654 

123 

9 

511750 

121 

9 

519697 

118 

9 

527501 

117 

3190 

503791 

3250 

511883 

3310 

519828 

3370 

527630 

1 

503927 

14 

1 

512017 

13 

1 

519959 

13 

1 

527759 

13 

2 

504063 

27 

2 

512150 

27 

2 

520090 

26 

2 

527888 

26 

3 

504199 

41 

3 

512284 

40 

3 

520221 

39 

3 

528016 

38 

4 

504335 

54 

4 

512417 

53 

4 

520352 

52 

4 

528145 

51 

5 

504471 

68 

5 

512551 

67 

5 

520483 

66 

5 

528274 

64 

6 

504607 

82 

6 

512684 

80 

6 

520614 

79 

6 

528402 

77 

7 

504743 

95 

7 

512818 

93 

7 

520745 

92 

7 

528531 

90 

8 

504878 

109 

8 

512951 

107 

8 

520876 

105 

8 

528660 

103 

9 

505014 

122 

9 

513084 

120 

9 

521007 

118 

9 

528788 

116 

3200 

505150 

3260 

513218 

3320 

521138 

3380 

528917 

1 

505286 

14 

1 

513351 

13 

1 

521269 

13 

1 

529045 

13 

2 

505421 

27 

2 

513484 

27 

2 

521400 

26 

2 

529174 

26 

3 

505557 

41 

3 

513617 

40 

3 

521530 

39 

3 

529302 

38 

4 

505692 

54 

4 

513750 

53 

4 

521661 

52 

4 

529430 

51 

5 

505828 

68 

5 

513883 

66 

5 

621792 

65 

5 

529559 

64 

6 

505963 

82 

6 

514016 

80 

6 

521922 

78 

6 

529687 

77 

7 

506099 

95 

7 

514149 

93 

7 

522053 

97 

7 

529815 

90 

8 

506234 

109 

8 

514282 

106 

8 

522183 

104  | 

8 

529943 

103 

9 

506370 

122 

9 

514415 

120 

9 

522314 

117 

9 

530072 

116 

3210 

506505 

3270 

514548 

3330 

522444 

3390 

530200 

1 

506640 

13 

1 

514680 

13 

1 

522575 

13 

1 

530328 

13 

2 

506775 

27 

2 

514813 

27 

2 

522705 

26 

2 

530456 

26 

3 

506911 

40 

3 

514946 

40 

3 

522835 

39 

3 

530584 

38 

4 

507046 

54 

4 

515079 

53 

4 

522966 

52 

4 

530712 

61 

5 

507181 

67 

5 

515211 

66 

5 

523096 

65 

5 

530840 

64 

6 

507316 

81 

6 

516344 

80 

6 

523226 

78 

6 

530968 

77 

7 

507451 

94 

7 

515476 

93 

7 

523356 

97 

7 

531095 

90 

8 

507586 

108 

8 

515609 

106 

8 

523486 

104  1 

8 

531223 

102 

9 

507721 

121 

9 

515741 

120 

9 

523616 

117  ! 

9 

531351 

115 

LOGARITHMS    OF   NUMBERS. 


13 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

3400 

531479 

3460  539076 

3520 

546543 

3580 

553883 

1 

531607 

13 

1  539202 

13 

1 

546666 

12 

1 

554004 

12 

2 

531734 

25 

2 

539327 

25 

2 

546789 

25 

2 

554126 

24 

3 

531862 

38 

3 

539452 

38 

3 

546913 

37 

3 

554247 

36 

4 

531990 

51 

4 

539578 

50 

4 

547036 

49 

4 

554368 

49 

5 

532117 

63 

5 

539703 

63 

5 

547159 

62 

5 

554489 

61 

6 

532245 

76 

6 

539829 

75 

6 

547282 

74 

6 

554610 

73 

7 

532372 

89 

7 

539954 

88 

7 

547405 

86 

7 

554731 

85 

8 

532500 

102 

8 

540079 

100 

8 

547529 

99 

8 

554852 

97 

9 

532627 

114 

9 

540204 

113 

9 

547652 

111 

9 

554973 

109 

3410 

532754 

3470 

540329 

3530 

547775 

3590 

555094 

1 

532882 

13 

1 

540455 

12 

1 

547898 

12 

1 

555215 

12 

2 

533009 

25 

2 

540580 

25 

2 

548021 

25 

2 

555336 

24 

3 

533136 

38 

3 

540705 

37 

3 

548144 

37 

3 

555457 

36 

4 

533263 

51 

4 

540830 

50 

4 

548266 

49 

4 

555578 

48 

5 

533391 

63 

5 

540955 

62 

5 

548389 

61 

5 

555699 

60 

6 

533518 

76 

6 

541080 

75 

6 

548512 

74 

6 

555820 

72 

7 

533645 

89 

7 

541205 

87 

7 

548635 

86 

7 

555.940 

84 

8 

533772 

102 

8 

541330 

100 

8 

548758 

98 

8 

556061 

96 

9 

533899 

114 

9 

541454 

112 

9 

548881 

111 

9 

556182 

108 

3420 

534026 

3480 

541579 

3540 

549003 

3600 

556302 

1 

534153 

13 

1 

541704 

12 

1 

549126 

12 

1 

556423 

,12 

2 

534280 

25 

2 

541829 

25 

2 

549249 

25 

2 

556544 

24 

3 

534407 

38 

3 

541953 

37 

3 

549371 

37 

3 

556664 

36 

4 

534534 

51 

4 

542078 

50 

4 

549494 

49 

4 

556785 

48 

5 

534661 

63 

5 

542203 

62 

5 

549616 

61 

5 

556905 

60 

6 

534787 

76 

6 

542327 

75 

6 

549739 

74 

6 

55702G 

72 

7 

534914 

89 

7 

542452 

87 

7 

549861 

86 

7 

557146 

84 

8 

535041 

102 

8 

542576 

100 

8 

549984 

98 

8 

557267 

96 

9 

535167 

114 

9 

542701 

112 

9 

550106 

111 

9 

557387 

108 

3430 

535294 

3490 

542825 

3550 

550228 

3610 

557507 

1 

535421 

13 

1 

542950 

12 

1 

550351 

12 

1 

557627 

12 

2 

535547 

25 

2 

543074 

25 

2 

550473 

24 

2 

557748 

24 

3 

535674 

38 

3 

543199 

37 

3 

550595 

37 

3 

557868 

36 

4 

535800 

50 

4 

543823 

50 

4 

550717 

49 

4 

557988 

48 

5 

535927 

63 

5 

543447 

62 

5 

550840 

61 

5 

558108 

60 

6 

536053 

76 

6 

543571 

75 

6 

550962 

73 

6 

558228 

72 

7 

536179 

88 

7 

543696 

87 

7 

'551084 

86 

7 

558348 

84 

8 

536306 

101 

8 

543820 

100 

8 

551206 

98 

8 

558469 

96 

9 

536432 

114 

9 

543944 

112 

9 

551328 

110 

9 

558589 

108 

3440 

536558 

3500 

544068 

3560 

551450 

3620 

558709 

1 

536685 

13 

1 

544192 

12 

1 

551572 

12 

1 

558829 

12 

2 

536811 

25 

2 

544316 

25 

2 

551694 

24 

2 

558948 

24 

3 

536937 

38 

3 

544440 

37 

3 

551816 

37 

3 

559068 

36 

4 

537063 

50 

4 

544564 

50 

4 

551938 

49 

4 

559188 

48 

5 

537189 

63 

5 

544688 

62 

5 

552059 

61 

5 

559308 

60 

6 

537315 

76 

6 

544812 

74 

6 

552181 

73 

6 

559428 

72 

7 

537441 

88 

7 

544936 

87 

7 

552303 

86 

7 

559548 

84 

8 

537567 

101 

8 

545060 

99 

8 

552425 

98 

8 

559667 

96 

9 

537693 

114 

9 

545183 

112 

9 

552546 

110 

9 

559787 

108 

3450 

537819 

3510 

545307 

3570 

552668 

3630 

559907 

1 

537945 

13 

1 

545431 

12 

1 

552790 

12 

1 

560026 

12 

2 

538071 

25 

2 

545554 

25 

2 

552911 

24 

2 

560146 

24 

3 

538197 

38 

3  545678 

37 

3 

553033 

36 

3 

560265 

36 

4 

538322 

50 

4 

545802 

49 

4 

553154 

49 

4 

560385 

48 

5 

538448 

63 

5 

545925 

62 

5 

553276 

61 

5 

560504 

60 

6 

538574 

76 

6 

546049 

74 

6 

553398 

73 

6 

560624 

72 

7 

538699 

88 

7  546172 

86 

7 

553519 

85 

7 

560743 

84 

8 

538825 

101 

8  546296 

99 

8 

553640 

97 

8 

560863 

96 

9 

538951 

114 

9  546419 

111 

9 

553762 

109 

9 

560982 

108 

14 


LOGARITHMS    OF   NUMBERS. 


NO. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

3640 

561101 

3700 

568202 

3760 

575188 

3820 

582063 

1 

561221 

12 

1 

568319 

12 

1 

575303 

12 

1 

582177  1  11 

2 

561340  24 

2 

568436 

23 

2 

575419 

23 

2 

582291  1  23 

3 

561459 

36 

3 

568554 

35 

3 

575534 

35 

3 

582404   34 

4 

561578 

48 

4 

568671 

47 

4 

575650 

46 

4 

582518  45 

5 

561698 

60 

5 

568788 

58 

5 

575765 

58 

5 

582631   56 

6 

561817 

72 

6 

568905 

70 

6 

575880 

69 

6 

582745   68 

7 

561936 

84 

7 

569023 

82 

7 

575996 

80 

7 

582858  79 

8 

562055 

96 

8 

569140 

94 

8 

576111 

92 

8 

582972   90 

9 

562174 

108 

9 

569257 

106 

9 

576226 

104 

9 

583085  '  102 

3650 

562293 

3710 

569374 

3770 

576341 

3830 

583199 

1 

562412 

12 

1 

569491 

12 

1 

576457 

12 

1 

583312  11 

2 

562531 

24 

2 

569608 

23 

2 

576572 

23 

2 

583426  23 

3 

562650 

36 

3 

569725 

35 

3 

576687 

35 

3 

583539  j  34 

4 

562768 

48 

4 

569842 

47 

4 

576802 

46 

4 

583652  j  45 

5 

562887 

60 

5 

569959 

58 

5 

576917 

58 

5 

583765  56 

6 

563006 

71 

6 

570076 

70 

6 

577032 

69 

6 

583879   68 

7 

563125 

83 

7 

570193 

82 

7 

577147 

80 

7 

583992  !  79 

8 

563244 

95 

8 

570309 

94 

8 

577262 

92 

8 

584105  90 

9 

563362 

107 

9 

570426 

106 

9 

577377 

104 

9 

584218  102 

3660 

563481 

3720 

570543 

3780 

577492 

3840 

584331  ; 

J 

563600 

12 

1 

570660 

12 

1 

577607 

11 

1 

584444   11 

2 

563718 

24 

i> 

570776 

23 

21577721 

23 

2  584557  !  23 

3 

563837 

36 

3 

570893 

35 

3  |  577836 

34 

3  584670  34 

4 

563955 

48 

4 

571010 

47 

4  577951 

46 

4  !  584783  45 

5 

564074 

60 

5 

571126 

58 

5  578066  57 

5  584896  '  56 

6 

564192 

71 

6 

571243 

70 

6 

578181 

68 

6  585009  :  68 

7 

564311 

83 

7 

571359 

81 

7 

578295 

80    7  585122  79 

8 

564429 

95 

8 

571476 

93 

8 

578410 

91  j    8  585235  ;  'JO 

9 

564548 

107 

9 

571592 

105 

9 

578525 

103  1   9  585348  102 

3670 

564666 

3730 

571709 

3790 

578639 

3850  585461  ! 

1 

564784 

12 

1 

571825 

12 

1 

578754 

11 

1 

585574 

11 

2 

564903 

24 

2 

571942 

:  23 

2 

578868 

23 

2 

585686 

22 

3 

565021 

36 

3 

572058 

35 

3 

578983 

34 

3 

585799 

34 

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655523 

38 

4 

661245 

38 

5 

643946 

49 

5 

649821 

49 

5 

655619 

48 

5 

661339 

47 

6 

644044 

59 

6 

649919 

58 

6 

655714 

58 

6 

661434 

57 

;  7 

644143 

69 

7 

650016 

68 

7 

655810 

67 

7 

661529 

66 

8 

644242 

79 

8 

650113 

78 

8 

655906 

77 

8 

661623 

76 

9 

644340 

89 

9 

650210 

88 

9 

656002 

86 

9 

661718 

85 

4410 

644439 

4470 

650307 

4530 

656098 

j!4590 

661813 

1 

644537 

10 

1  650405 

10 

1 

656194 

10 

1 

661907 

9 

2 

644635 

20 

2  650502 

19 

2 

656290 

19 

2 

662002 

19 

3 

644734 

30 

3 

650599 

29 

3 

656386 

29 

3 

662096 

28 

4 

644832 

39 

4 

650696 

39 

4 

656481 

38 

4 

662191 

38 

5 

644931 

49 

5 

650793 

49 

5 

656577 

48 

5 

662285 

47 

6 

645029 

59 

6 

650890 

58 

6 

656673 

58 

6 

662380 

57 

7 

645127 

69 

7  650987 

68 

7 

656769 

67 

7 

662474 

66 

8  645226 

79 

8  651084 

78 

8 

656864 

77 

8 

662569 

76 

9  645324 

89 

9  651181 

88 

9 

656960 

86. 

9 

662663 

85 

18 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Pare. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part, 

4600 

662758 

4660 

668386 

4720 

673942 

4780 

679428 

1 

662852 

9 

1 

668479 

9 

1 

674034 

9 

1 

679519 

9 

2 

662947 

19 

2 

668572 

19 

2 

674126 

18 

2 

679610 

18- 

3 

663041 

28 

3 

668665 

28 

3 

674218 

28 

3 

679700 

27 

4 

663135 

38 

4 

668758 

37 

4 

674310 

37 

4 

679791 

36 

5 

663230 

47 

5 

668852 

47 

5 

674402 

46 

5 

679882 

45 

6 

663324 

57 

6 

668945 

56 

6 

674494 

55 

6 

679973 

55 

7 

663418 

66 

7 

669038 

65 

7 

674586 

64 

7 

680063 

64 

8 

663512 

76 

8 

669131 

74 

8 

674677 

74 

8 

680154 

73 

9 

663607 

85 

9 

669224 

84 

9 

674769 

83 

9 

680245 

82 

4610 

663701 

4670 

669317 

4730 

674861 

4790 

680335 

1 

663795 

9 

1 

669410 

9 

1 

674953 

9 

1 

680426 

9 

2 

663889 

19 

2 

669503 

19 

2 

675045 

18 

2 

680517 

18 

3 

663983 

28 

3 

669596 

28 

3 

675136 

28 

3 

680607 

27 

4 

664078 

38 

4 

669689 

37 

4 

675228 

37 

4 

680698 

36 

5 

664172 

47 

5 

669782 

47 

5 

675320 

46 

5 

680789 

45 

6 

664266 

56 

6 

669875 

56 

6 

675412 

55 

6 

680879 

55 

7 

664360 

66 

7 

669967 

65 

7 

675503 

64 

7 

680970 

64 

8 

664454 

75 

8 

670060 

74 

8 

675595 

74 

8 

681060 

73 

9 

664548 

85 

9 

670153 

84 

9 

675687 

83 

9 

681151 

82 

4620 

664642 

4680 

670246 

4740 

675778 

4800 

681241 

1 

664736 

9 

1 

670339 

9 

1 

675870 

9 

1 

681332 

9 

2 

664830 

19 

2 

670431 

18 

2 

675962 

18 

2 

681422 

18 

3 

664924 

28 

.3 

670524 

28 

3 

676053 

27 

3 

681513 

27 

4 

665018 

38 

4 

670617 

37 

4 

676145 

36 

4 

681603 

36 

5 

665112 

47 

5 

670710 

46 

5 

676236 

46 

5 

681693 

4o 

6 

665206 

56 

6 

67UJS02 

55 

6 

676328 

55 

6 

681784 

54 

7 

665299 

66 

7 

670895 

64 

7 

676419 

64 

7 

681874 

63 

8 

665393 

75 

8, 

670988 

74 

8 

676511 

73 

8 

681964 

72 

9 

665487 

85 

9 

671080 

83 

9 

676602 

82 

9 

682055 

81  ! 

4630 

665581 

4690 

671173 

4750 

676694 

4810 

682145 

1 

665675 

9 

1 

671265 

9 

1 

676785 

9 

1 

682235 

9 

2 

665769 

19 

2 

671358 

18 

2 

676876 

18 

2 

682326 

18 

3 

665862 

28 

3 

671451 

28 

3  676968 

27 

3 

682416 

27 

4 

665956 

38 

4 

671543  37 

4  1  677059 

36 

4 

682506 

36 

5 

666050 

47 

5 

671636  46 

5 

677151 

46 

5 

682596 

45 

6 

666143 

56 

6 

671728 

55 

6 

677242 

55 

6 

682686 

'54 

7 

666237 

66 

7 

671821 

64 

7 

677333 

64 

7 

682777 

63 

8 

666331 

75 

8 

671913 

74 

8 

677424 

73 

8 

682867 

72 

9 

666424 

85 

9 

672005 

83 

9 

677516 

82 

9 

682957 

81 

4640 

666518 

4700 

672098 

4760 

677607 

4820 

683047 

1 

666612 

9 

1 

672190 

9 

1 

677698 

9 

1 

683137 

9 

2 

666705 

19 

•> 

672283 

18 

2 

677789 

18 

2 

683227 

18 

3  |  666799 

28 

3 

672375 

28 

3 

677881 

27 

3 

683317 

27 

4 

666892 

37 

4 

672467 

37 

4 

677972 

36 

4 

683407 

36 

5 

666986 

47 

5 

672560 

46 

5 

678063 

45 

5 

683497 

45 

6 

667079 

56 

6 

672652 

55 

6 

678154 

65 

6 

683587 

54 

,7 

667173 

65 

7 

672744 

64 

7 

678245 

64 

7 

683677 

63 

8 

667266 

74 

8 

672836 

74 

8 

678336 

73 

8 

683767 

72 

9  i  607359 

84 

9 

672929 

83 

9 

678427 

82 

9 

683857 

81 

4650  667453 

4710 

673021 

4770 

678518 

4830 

683947 

1  667546 

9 

1 

673113 

9 

1 

678609 

9 

1 

684037 

9 

2  |  667640 

19 

2 

673205 

18 

2 

678700 

18 

2 

684127 

18 

3  i  667733 

28 

3 

673297 

28 

3 

678791 

27 

3 

684217 

27 

4  I  667826 

37 

4 

673390 

37 

4 

678882 

36 

4 

684307 

36 

5  1  667920 

47 

5 

673482 

46 

5 

678973 

45 

5 

684396 

45 

6 

668013 

56 

6 

673574 

55 

6 

679064 

55 

6 

684486 

54 

7 

668106 

65 

7 

673666 

64 

7 

679155 

64 

7 

684576 

63 

8 

668199 

74 

8 

673758 

74 

8 

679246 

73 

8 

684666 

72 

9 

668293 

84 

9 

673850 

83     9 

679337 

82 

9 

684756 

81 

LOGARITHMS    OF    NUMBERS. 


19 


No. 

Log. 

/roj'. 
fttr.. 

No. 

I**. 

Prop. 
Fare. 

No.    Log.   !££ 

No. 

Log. 

i'iop.  ; 

JJurt. 

4840 

684845 

4900 

690196 

i 
4960 

695482 

5020 

700704 

1 

684935 

9 

1 

690285 

9 

1 

695569 

9 

1 

700790 

9 

2 

685025 

18 

2 

690373 

18 

2 

695657 

17 

2 

700877 

17 

3 

685114 

27 

3 

690462 

27 

3 

695744 

26 

3 

700963 

26 

4 

685204 

36 

4 

690550 

35 

4 

695832 

35 

4  701050 

35 

5 

685294 

45 

5 

690639 

44 

5 

695919 

44 

5 

701136 

43 

6 

685383 

54 

6 

690727 

53 

6 

696007 

52 

6 

701222 

52 

7 

685473 

63 

7 

690816 

62 

7 

696094 

61 

7 

701309 

61 

8 

685563 

72 

8 

690905 

71 

8 

696182 

70 

8 

701395 

70 

9 

685652 

81 

9 

690993 

80 

9 

696269 

79 

9 

701482 

78 

4850 

685742 

4910 

691081 

4970 

696356 

5030 

701568 

1 

685831 

9 

1 

691170 

9 

1 

696444 

9 

1 

701654 

9  ! 

2 

6«5921 

18 

2 

691258 

18 

2 

696531 

17 

2 

701741 

17 

3 

686010 

27 

3 

691347 

27 

3 

696618 

26 

3 

701827 

26 

4 

686100 

36 

4 

691435 

35 

4 

696706 

35 

4 

701913 

35 

5 

686189 

45 

5 

691524 

44 

5 

696793 

44 

5 

701999 

43 

6 

686279 

54 

6 

691612 

53 

6 

696880 

52 

6 

702086 

52 

7 

686368 

63 

7 

691700 

62 

7 

696968 

61 

7 

702172 

61 

8 

686457 

72 

8 

691789 

71 

8 

697055 

70 

8 

702258 

70 

9 

686547 

81 

9 

691877 

80 

9 

697142 

79 

9 

702344 

78 

4860 

686636 

4920 

691965 

4980 

697229 

5040 

702430 

1  1  686726 

9 

1 

692053 

9 

1 

697317 

9 

1 

702517 

9 

2 

686815 

18 

2 

692142 

18 

2 

697404 

17 

2 

702603 

17 

3 

686904 

27 

3 

692230 

27 

3 

697491 

26 

3 

702689 

26 

4 

686994 

36 

4 

692318 

35 

4  1  697578 

35 

4 

702775 

34 

5 

687083 

45 

5 

692406 

44 

5 

697665 

44 

5 

702861 

43 

6 

687172 

54 

6 

692494 

53 

6 

697752 

52 

6 

702947  |  52 

7 

687261 

63 

7 

692583 

62 

7 

697839 

61 

7 

703033 

60 

8 

687351 

72 

8 

692671 

71 

8 

697926 

70 

8 

703119 

69 

9 

687440 

81 

9 

692759 

80 

9 

698013 

79 

9 

703205 

77 

4870 

687529 

•4930 

692847 

4990 

698100 

5050 

703291 

1 

687618 

9 

1 

692935 

9 

1 

698188 

9 

1 

703377 

9 

2 

687707 

18 

2 

693023 

18 

2 

698275 

17 

2 

703463 

17 

3 

687796 

27 

3 

693111 

26 

3 

698362 

26 

3 

703549 

26 

4 

687886 

36 

4 

693199 

35 

4 

698448 

35 

4 

703635 

34 

5 

687975 

45 

5 

693287 

44 

5 

698535 

44 

5 

703721 

43 

6 

688064 

54 

6 

693375 

53 

6 

698622 

52 

6 

703807 

52 

7 

688153 

62 

7 

693463 

62 

7 

698709 

61 

7 

703893 

60 

8 

688242 

72 

8 

693551 

70 

8 

698796 

70 

8 

703979 

69 

9 

688331 

80 

9 

693639 

79 

9 

698883 

79 

9 

704065 

77 

4880 

688420 

4940 

693727 

5000 

698970 

5060 

704150 

1 

688509 

9 

1 

693815 

9 

1 

699057 

9 

1 

704236 

9 

2 

688598 

18 

2 

693903 

18 

2  i  699144 

17 

2 

704322 

17 

3 

688687 

27 

3 

693991 

26 

3  j  699231 

26 

3 

704408 

26 

4 

688776 

36 

4 

694078 

35 

4  699317 

35 

4 

704494 

34 

5 

688865 

45 

5 

694166 

44 

5 

699404 

43 

5 

704579 

43 

6 

688953 

54 

6 

694254 

53 

6 

699491 

52 

6 

704665 

52 

7 

689042 

62 

7 

694342 

62 

7 

699578 

61 

7 

704751 

60 

'  8 

689131 

72 

8 

694430 

70 

8 

699664 

70 

8 

704837 

69 

9 

689220 

80 

9 

694517 

79 

9 

699751 

78 

9 

704922 

77 

4890 

689309 

4950 

694605 

5010 

699838 

5070 

705008 

1  |  689398 

9 

1 

69469-3 

9 

1 

699924 

9 

1 

705094 

9 

2 

689486 

18 

2 

694781 

18 

2 

700011 

17 

2  705179 

17 

3 

689575 

27 

g 

694868 

26 

3 

700098 

26 

3  705265 

26 

4 

689664 

36 

4 

694956 

35 

4 

700184 

35 

4  705350 

34 

5 

G89753 

45 

- 

695044 

44 

5 

700271 

43 

5 

705436 

43 

6 

689841 

54 

6 

695131 

53 

6 

700358 

52 

6 

705522 

52 

7 

689930 

62 

7 

695219 

62 

7 

700444 

61 

7  705607 

60 

8 

690019 

72 

8 

695307 

70 

8 

700531 

70 

8  j  705693 

69 

9 

690107 

80 

9 

(395394 

79 

9 

700617 

78 

9  1  705778 

77 

20 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop.  ;   M 
Part,  i   No- 

Log. 

S:  NO. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

5080 

705864 

5140 

710963 

5200 

716003 

5260 

720986 

1 

705949 

9 

1 

711048 

8  ,   1 

716087 

8 

1 

721068 

8 

2 

706035 

17 

2 

711132 

17 

2 

716170 

17 

2 

721151 

16 

3 

706120 

26 

3 

711216 

25 

3 

716254 

25 

3 

721233 

25 

4 

706206 

34 

4 

711301 

34 

4 

716337 

34 

4 

721316 

33 

5 

706291 

43 

5 

711385 

42 

5 

716421 

42 

5 

721398 

41 

6 

706376 

51 

6 

711470 

51 

6 

716504 

50 

6 

721481 

49 

7 

706462 

60 

7 

711554 

59 

7 

716588 

59 

7 

721563 

58 

8 

706547 

68 

8 

711638 

68 

8 

716671 

-67 

8 

721646 

66 

9 

706632 

77 

9 

711723 

76 

9 

716754 

76 

9 

721728 

74 

5090 

706718 

5150 

711807 

5210 

716838 

5270 

721811 

1 

706803 

Q 

1 

711892 

8 

1 

716921 

8 

1 

721893 

8 

2 

706888 

17 

2 

711976 

17  ; 

2 

717004 

17 

2 

721975 

16 

3 

706974 

26 

3 

712060 

25 

3 

717088 

25 

3 

722058 

25 

4 

707059 

34 

4 

712144 

34  ! 

4 

717171 

33 

4 

722140 

33 

5 

707144 

43 

5 

712229 

42 

5 

717254 

42 

5 

722222 

41 

6 

707229 

51 

6 

712313 

51  | 

6 

717338 

50 

6 

722305 

49 

7 

707315 

60 

7 

712397 

59 

7 

717421 

58 

7 

722387 

58 

8 

707400 

68 

8 

712481 

68 

8 

717504 

66 

8 

722469 

66 

9 

707486 

77 

9 

712566 

76 

9 

717587 

75 

9 

722552 

74 

5100 

707570 

15160 

712650 

5220 

717671 

5280 

722634 

1 

707655 

9 

1 

712734 

8 

1 

717754 

8 

1 

722716 

8 

2 

707740 

17 

2 

712818 

17 

2 

717837 

17 

2 

722798 

16 

3 

707826 

26 

3 

712902 

25 

3 

717920 

25 

3 

722881 

25 

4 

707911 

34 

4 

712986 

34 

4 

718003 

33 

4 

722963 

33 

5 

707996 

43 

5 

713070 

42 

5 

718086 

42 

5 

723045 

41 

(i 

708081 

51 

6 

713154 

50 

6 

718169 

50 

6 

723127 

49 

7 

708166 

60     7 

713238 

59 

7 

718253 

58 

7 

723209 

58 

8  708251 

68 

8 

713322 

67 

8 

718336 

66 

8 

723291 

66 

9  i  708336 

77 

9 

713406 

76 

9 

718419 

75 

9 

723374 

74 

5110  708421 

5170 

713490 

5230 

718502 

5290 

723456 

1  708506 

9 

'1 

713574 

8 

1 

718585 

8 

1 

723538 

8 

2  :  708591 

17 

2 

713658 

17 

2 

718668 

17 

2  723620 

16 

3  708676 

26 

3 

713742 

25 

3 

718751 

25 

3  |  723702 

25 

4  708761 

34 

4 

713826 

34 

4 

718834 

33 

4  :  723784 

33 

5  708846 

43 

5 

713910 

42 

5 

718917 

42 

5  i  723866 

41 

6  :  708931 

51 

6 

713994 

50 

6 

719000 

50 

6  |  723948 

49 

7  !  709015 

60 

7 

714078 

59 

7 

719083 

58 

7 

724030 

57 

8  I  709100 

68 

8 

714162 

67 

8 

719165 

66 

8 

724112 

66 

9  709185 

77 

9 

714246 

76 

9 

719248 

75 

9 

724194 

74 

5120  i  709270 

5180 

714330 

5240 

719331 

5300 

724276 

1  709355 

8 

1 

714414 

8 

1 

719414 

8 

1 

724358 

8 

2  709440 

17 

.  2 

714497 

17 

2 

719497 

17 

2 

724440 

16 

3  709524 

25 

3 

714581 

25 

3 

719580 

25 

3 

724522 

25 

4  709609 

34 

4 

714665 

34 

4 

719663 

33 

4 

724603 

33 

5  ;  709694 

42 

5 

714749 

42 

5 

719745 

41 

5 

724685 

41 

6  709779 

51 

6 

714832 

50 

6 

719828 

50 

6 

724767 

49 

7  !  709863 

59 

7 

714916 

59 

7 

719911 

58 

7 

724849 

57 

8  709948 

68 

8 

715000 

67 

,8 

719994 

66 

8 

724931 

613 

9  710033 

76 

9 

715084 

76 

9 

720077 

75 

9 

725013 

74 

5130  710117 

5190 

715167 

5250 

720159 

5310 

725095 

1  710202 

8 

1 

715251 

8 

1 

720242 

8 

1 

725176 

8 

2  710287 

17 

2 

715335 

17 

2 

720325 

17 

2 

725258 

16 

3  710371 

25 

3 

715418 

25 

3 

720407 

25 

3 

725340 

25 

4  710456 

34 

4  715502 

34 

4 

720490 

33 

4 

72542-2 

33 

5  710540 

42 

5  715586 

42 

5' 

720573 

41 

5 

725503 

41 

6  710625 

51 

6  I  715669 

50 

6 

720655 

50 

6 

725585 

49 

7  710710 

59 

7,  715753 

59 

7 

720738 

58 

7 

725667 

57 

8  710794 

67 

8  i  715836 

67 

8' 

720821 

66 

8 

725748 

tJ6 

9  710879 

76 

9  715920  76 

9 

720903 

75 

9 

725830 

74 

LOGARITHMS    OF   NUMBERS. 


21 


No. 

Log. 

Prop. 
Part. 

i 

No. 

**   ?a°r?: 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

5320 

725912 

5380 

730782 

5440 

735599 

5500 

740363 

1 

725993 

8 

1 

730863 

8 

1 

735679 

8 

1 

740442 

8 

2 

726075 

16 

2 

730944 

16 

2 

735759 

16 

2 

740521 

16 

3 

726156 

24 

3 

731024 

24 

3 

735838 

24 

3 

740599 

24 

4 

726238 

33 

4 

731105 

32 

4 

735918 

32 

4 

74067"8 

32 

5 

726320 

41 

5 

731186 

40 

5 

735998 

40 

5 

740757 

40 

6 

726401 

49 

6 

731266 

49 

6 

736078 

48 

6 

740836 

47 

7 

726483 

57 

7 

731347 

57 

7 

736157 

56 

7 

740915 

55 

8 

726564 

65 

8 

731428 

65 

8 

736237 

64 

8 

740994 

63 

9 

726646 

73 

9 

731508 

73 

9 

736317 

72 

9 

741073 

71 

5330 

726727 

5390 

731589 

5450 

736396 

5510 

741152 

1 

726809 

8 

1 

731669 

8 

1 

736476 

8 

1 

741230 

8 

2 

726890 

16 

2 

731750 

16 

2 

736556 

16 

2 

741309 

16 

3 

726972 

24 

3 

731830 

24 

3 

736635 

24 

3 

741388 

24 

4 

727053 

33 

4 

731911 

32 

4 

736715 

32 

4 

741467 

32 

5 

727134 

41 

5 

731991 

40 

5 

736795 

40 

5 

741546 

40 

6 

727216 

49 

6 

732072 

48  1 

6 

736874 

48 

6 

741624 

47 

7 

727297 

57 

7 

732152 

56 

7 

736954 

56 

7 

741703 

55 

8 

727379 

65 

8 

732233 

64 

8 

7-37034 

64 

8 

741782 

63 

9 

727460 

73 

9 

732313 

72 

9 

737113 

72 

9 

741860 

71 

5340 

727541 

5400 

732394 

5460 

737193 

5520 

741939 

1 

727623 

8 

1 

732474 

8 

1 

737272 

8 

1 

742018 

8 

2 

727704 

16 

2 

732555 

16 

2 

737352 

16 

2 

742096 

16 

3 

727785 

24 

3 

732635 

24 

3 

737431 

24 

3 

742175 

23 

4 

727866 

33 

4 

732715 

32 

4 

737511 

32 

4 

742254 

31 

5 

727948 

41 

5 

732796 

40 

5 

737590 

40 

6 

742332 

39 

6 

728029 

49 

6 

732876 

48 

"6 

737670 

48 

6 

742411 

47 

7 

728110 

57 

7 

732956 

56 

7 

737749 

56 

7 

742489 

55 

8 

728191 

65 

8 

733037 

64 

8 

737829 

64 

8 

742568 

63 

9 

728273 

73 

9 

733117 

72 

9 

737908 

72 

9 

742647 

71 

5350 

728354 

5410 

733197 

5470 

737987 

5530 

742725 

1 

728435 

8 

1 

733278 

8 

1 

738067 

8 

1 

742804 

8 

2 

728516 

16 

2 

733358 

16     2 

738146 

16 

2 

742882 

16 

3 

728597 

24 

3 

733.438 

24 

3 

738225 

24 

3 

742961 

23 

4 

728678 

33 

4 

733518 

32 

4 

738305 

32 

4 

743039 

31 

5 

728759 

41 

5 

733598 

40  |    5 

738384 

40 

5  743118 

39 

6 

728841 

49 

6 

733679 

48 

6 

738463 

48 

6  743196 

47 

7 

728922 

57 

7 

733759 

56 

7 

738543 

56 

7  743275 

55 

8 

729003 

65 

8 

733839 

64 

8 

738622 

64 

8  743353 

63 

9 

729084 

73 

9 

733919 

72 

9 

738701 

72 

9 

743431 

71 

5360 

729165 

5420 

733999 

5480 

738781 

5540 

743510 

1 

729246 

8  <    1 

734079 

8 

1 

738860 

8 

1 

743588 

8 

2 

729327 

16  !   2 

734159 

16 

2 

738939 

16 

2 

743667 

16 

3 

729408 

24 

3 

734240 

24 

3 

739018 

24 

3 

743745 

23 

4 

729489 

32 

4 

734320 

32 

4 

739097 

32 

4 

743823 

31 

5 

729570 

41 

5 

734400 

40 

5 

739177 

40 

5 

743902 

39 

6 

729651 

49 

6 

734480 

48 

6 

739256 

47 

6 

743980 

47 

7 

729732 

57 

7 

734560 

56 

7 

739335 

55 

7 

744058 

55 

8 

729813 

65 

8 

734640 

64 

8 

739414 

63 

8 

744136 

63 

9 

729893 

73 

9 

734720 

72 

9 

739493 

71 

9 

744215 

71 

5370 

729974 

6430 

734800 

5490 

739572 

5550 

744293 

1 

730055 

8 

1 

734880 

8 

1 

739651 

8 

1 

744371 

8 

2 

730136 

16 

2 

734960 

16 

2 

739730 

16 

2 

744449 

16 

3 

730217 

24 

3 

735040 

24 

8 

739810 

24 

3 

744528 

23 

4 

730298 

32 

4 

735120 

32 

4 

739889 

32  j   4 

744606 

31 

5 

730378 

40 

5 

735200 

40 

5 

739968 

40 

5 

744684 

39 

6 

730459 

49 

6 

735279 

48 

6 

740047 

47 

6 

744762 

47 

7 

730540 

57 

7 

735359 

56 

7 

740126 

55 

7 

744840 

65 

8 

730621 

65 

8 

735439 

64 

8 

740205 

63 

8 

744919 

63 

9 

730702 

73 

9 

735519 

72 

9 

740284 

71 

9 

744997 

71 

LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

L«s-  IK; 

5560 

745075 

5620 

749736 

5680 

754348 

5740 

758912 

1 

745153 

8 

1 

749814 

8 

1 

754425 

8 

1 

758988 

8 

2 

745231 

16 

2 

749891 

16 

2 

754501 

15 

2 

759063 

15 

8 

745309 

23 

3 

749968 

23 

3 

754578 

23 

3 

759139 

23 

4 

745387 

31 

4 

750045 

31 

4 

754654 

30 

4 

759214 

30 

5 

745465 

39 

5 

750123 

39 

5 

754730 

38 

5 

759290 

38 

6 

745543 

47 

6 

750200 

47 

6 

754807 

46 

6 

759366 

45 

7 

745621 

55 

7 

750277 

54 

7 

754883 

53 

7 

759441 

53 

8 

745699 

62 

8 

750354 

62 

8 

754960 

61 

8 

759517 

60 

9 

745777 

70 

9 

750431 

70 

9 

755036 

69 

9 

759592 

68 

5570 

745855 

5630 

750508 

5690 

755112 

5750 

759668 

i 

1 

745933 

8 

1 

750586 

8 

1 

755189 

8 

1 

759743 

8 

2 

746011 

16 

2 

750663 

16 

2 

755265 

15 

2 

759819 

15 

3 

746089 

23 

3 

750740 

23 

3 

755341 

23 

3 

759894 

23 

4 

746167 

31 

4 

750817 

31 

4 

755417 

3.0 

4 

759970 

30 

5 

746245 

39 

5 

750894 

39 

5 

755494 

38 

5 

760045 

38 

6 

746323 

47 

6 

750971 

47 

6 

755570 

46 

6 

760121 

45 

7 

746401 

55 

7 

751048 

54 

7 

755646 

53 

7 

760196 

53 

8 

746479 

62 

8 

751125 

62 

8 

755722 

61 

8 

760272 

60 

9 

746556 

70 

9 

751202 

70 

9 

755799 

69 

9 

760347 

68 

5580 

746634 

5640 

751279 

5700 

755875 

5760 

760422 

1 

746712 

8 

1 

751356 

8 

1 

755951 

8 

1 

760498 

8 

2 

746790 

16 

2 

751433 

15 

0 

756027 

15 

2 

760573 

15 

3 

746868 

23 

3 

751510 

23 

8 

756103 

23 

3 

760649 

23 

4 

746945 

31 

4 

751587 

30 

4 

756180 

30 

4 

760724 

30 

5 

747023 

39 

5 

751664 

38 

5 

756256 

38 

5 

760799 

38 

6 

747101 

47 

6 

751741 

46 

6 

756332 

46 

6 

760875 

45 

7 

747179 

55 

7 

751818 

54 

7 

756408 

53 

7 

760950 

53 

8 

747256 

62 

8 

751895 

62 

8 

756484 

61 

8 

761025 

60 

9 

747334 

70 

9 

751972 

70 

9 

756560 

69 

9 

761100 

68 

5590 

747412 

5650 

752048 

5710 

756636 

5770 

761176 

1 

747489 

8 

1 

752125 

8 

1 

756712 

8 

1 

761251 

8 

2 

747567 

16 

2 

752202 

15 

2 

756788 

15 

2 

761326 

15 

3 

747645 

23 

3 

752279 

23 

3 

756864 

23 

3 

761402 

23 

4 

747722 

31 

4 

752356 

30 

4 

756940 

30 

4 

761477 

30 

5 

747800 

39 

5 

752433 

38 

5 

757016 

38 

5 

761552 

38 

6 

747878 

47 

6 

752509 

46 

6 

757092 

46 

6 

761627 

45 

7 

747955 

54 

7 

752586 

54 

7 

757168 

53 

7 

761702 

53 

8 

748033 

62 

8 

752663 

62 

8 

757244 

61 

8 

761778 

60 

9 

748110 

70 

9 

752740 

70 

9 

757320 

69 

9 

761853 

68 

5600 

748188 

5660 

752816 

5720 

757396 

5780 

761928 

1 

748266 

8 

1 

752893 

8 

1 

757472 

8 

1 

762003 

8 

2 

748343 

16 

2 

752970 

15 

2 

757548 

15 

2 

763078 

15 

3 

748421 

23 

3 

753047 

23 

3 

757624 

23 

-  3 

762153 

22 

4 

748498 

31 

4 

753123 

30 

4 

757700 

30 

4 

762228 

30 

5 

748576 

39 

5 

753200 

38 

5 

757775 

38 

5 

762303 

38 

6 

748653 

47 

6 

753277 

46 

6 

757851 

46 

6 

762378 

45 

7 

748731 

54 

7 

753353 

54 

7 

757927 

sa 

7 

762453 

52 

8 

748808 

62 

8 

753430 

62 

8 

758003 

61 

8 

762529 

60 

9 

748885 

70 

9 

753506 

70 

9 

758079 

68 

9 

762604 

68 

5610 

748963 

5670 

753583 

5730 

758155 

5790 

762679 

1 

749040 

8 

1 

753660 

8 

1 

758230 

8 

1 

762754 

8 

2 

749118 

16 

2 

753736 

15 

2 

758306 

15 

2 

762829 

15 

3 

749195 

23 

3 

753813 

23 

3 

758382 

23 

3 

762904 

22 

4 

749272 

31 

4 

753889 

30 

4 

758458 

30 

4 

762978 

30 

5 

749350 

39 

5 

753966 

38 

5 

758533 

38 

5 

763053 

38 

6 

749427 

47 

6 

754042 

46 

G 

758609 

46 

6 

763128 

45 

7 

749504 

54 

7 

754119 

54 

7 

758685 

53 

7 

763203 

52 

8 

749582 

62 

8 

754195 

62 

8 

758760 

61 

8 

763278 

60 

9 

749659 

70 

9 

754272 

70 

9  758836 

68 

9 

763353 

68 

LOGARITHMS    OP   NUMBERS. 


23 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

5800 

703428 

5860 

767898 

5920 

772322 

5980 

776701 

1 

763503 

7 

1 

767972 

7 

1 

772395 

7 

1 

776774 

7 

2 

763578 

15 

2 

768046 

15 

2 

772468 

15 

2 

776846 

14 

3 

763653 

22 

3 

768120 

22 

3 

772542 

22 

3 

776919 

22 

4 

763727 

30 

4 

768194 

30 

4 

772615 

29 

4 

776992 

29 

5 

763802 

37 

5 

768268 

37 

5 

772688 

37 

5 

777064 

36 

6 

763877 

45 

6 

768342 

45 

6 

772762 

44 

6 

777137 

43  , 

7 

763952 

52 

7 

768416 

52 

7 

772835 

51 

7 

777209 

51 

8 

764027 

60 

8 

768490 

59 

8 

772908 

59 

8 

777282 

58 

9 

764101 

67 

9 

768564 

67 

9 

772981 

66 

9 

777354 

65 

5810 

764176 

5870 

768638 

5930 

773055 

5990 

777427 

1 

764251 

7 

1 

768712 

7 

1 

773128 

7 

1 

777499 

7 

2 

764326 

15 

2 

768786 

15 

2 

773201 

15 

2 

777572 

14 

3 

764400 

22 

8 

768860 

22 

3 

773274 

22 

3 

777644 

22 

4 

764475 

30 

4 

768934 

30 

4 

773348 

29 

4 

777717 

29 

5 

764550 

37 

5 

769008 

37 

5 

773421 

37 

5 

777789 

36 

6 

764624 

45 

6 

769082 

45 

6 

773494 

44 

6 

777862 

43 

7 

764699 

52 

7 

769156 

52 

7 

773567 

51 

7 

777934 

51 

8 

764774 

60 

8 

769230 

59 

8 

773640 

59 

8 

778006 

58 

9 

764848 

67 

9 

769303 

67 

9 

773713 

66 

9 

778079 

65 

5820 

764923 

5880 

769377 

5940 

773786 

6000 

778151 

1 

764998 

7 

1 

769451 

7 

1 

773860 

7 

1 

778224 

7 

2 

765072 

15 

2 

769525 

15 

2 

773933 

15 

2 

778296 

14 

3 

765147 

22 

3 

769599 

22 

3 

774006 

22 

3 

778368 

22 

4 

765221 

30 

4 

769673 

30 

4 

774079 

29 

4 

778441 

29 

5 

765296 

37 

5 

769746 

37 

5 

774152 

37 

5 

778513 

36 

6 

765370 

45 

6 

769820 

45 

6 

774225 

44 

6 

778586 

43 

7 

765445 

52 

7 

769894 

52 

7 

774298 

51 

7 

778658 

51 

8 

765520 

60 

8 

769968 

59 

8 

774371 

59 

8 

778730 

58 

9 

765594 

67 

9 

770042 

6v 

9 

774444 

66 

9 

778802 

65 

5830 

765669 

5890 

770115 

5950 

774517 

6010 

778874 

1 

765743 

7 

1 

770189 

7 

1 

774590 

7 

1 

778947 

7 

2 

765818 

15 

2 

770263 

15 

2 

774663 

15 

2 

779019 

14 

8 

765892 

22 

3 

770336 

22 

3 

774736 

22 

3 

779091 

22 

4 

765966 

30 

4 

770410 

30 

4 

774809 

29 

4 

779163 

29 

5 

766041 

37 

5 

770484 

37 

5 

774882 

37 

5 

779236 

36 

6 

766115 

45 

6 

770557 

45 

6 

774955 

44 

6 

779308 

43 

7 

766190 

52 

7 

770631 

52 

7 

775028 

51 

7 

779380 

51 

8 

766264 

60 

8 

770705 

59 

8 

775100 

59 

8 

779452 

58 

9 

766338 

67 

9 

770778 

67 

9 

775173 

66 

9 

779524 

65 

5840 

766413 

5900 

770852 

5960 

775246 

6020 

779596 

1 

766487 

7 

1 

770926 

7 

1 

775319 

7 

1 

779669 

7 

2 

766562 

15 

2 

770999 

15 

2 

775392 

15 

2 

779741 

14 

3 

766636 

22 

3 

771073 

22 

3 

775465 

22 

3 

779813 

22 

4 

766710 

30 

4 

771146 

30 

4 

775538 

29 

4 

779885 

29 

5 

766785 

37 

5 

771220 

37 

5 

775610 

37 

5 

779957 

36 

6 

766859 

45 

6 

771293 

45 

6 

775683 

44 

6 

780029 

43 

7 

766933 

52 

7 

771367 

52 

7 

775756 

51 

7 

780101 

50 

8 

767007 

60 

8 

771440 

59 

8 

775829 

•59 

8 

780173 

58 

9 

767082 

67 

9 

771514 

67 

9 

775902 

66 

9 

780245 

65 

5850 

767156 

5910 

771587 

5970 

775974 

6030 

780317 

1 

767230 

7 

1 

771661 

7 

1 

776047 

7 

1 

780389 

7 

2 

767304 

15 

2 

771734 

15 

2 

776120 

15 

2 

780461 

14 

3 

767379 

22 

3 

771808 

22 

3 

776193 

22 

3 

780533 

22 

4 

767453 

30 

4 

771881 

30 

4 

776265 

29 

4 

780605 

29 

5 

767527 

37 

5 

771955 

37 

5 

776338 

37 

5 

780677 

36 

6 

767601 

45 

6 

772028 

44 

6 

776411 

44 

6 

780749 

43 

7 

767675 

52 

7 

772102 

52 

7 

776483 

51 

7 

780821 

50 

8 

767749 

59 

8 

772175 

59 

8 

776556 

59 

8 

780893 

58 

9 

767823 

67 

9 

772248 

67 

9 

776629 

66 

9 

780965 

65J 

24 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

ss- 

No. 

Log. 

Prop.] 
Part,  i 

6040 

781037 

6100 

785330 

6160 

789581 

- 

6220 

793790 

1 

781109 

7 

1 

785401 

7 

1 

789651 

7 

1 

793860 

7 

2 

781181 

14 

2 

785472 

14 

2 

789722 

14 

2 

793930 

14 

3 

781253 

22 

3 

785543 

21 

3 

789792 

21 

3 

794000 

21 

4 

781324 

29 

4 

785615 

28 

4 

789863 

28 

4 

794070 

28 

5 

781396 

36 

5 

785686 

36 

5 

789933 

35 

5 

794139 

35 

6 

781468 

43 

6 

785757 

43 

6 

790004 

42 

6 

794209 

42 

•7 

781540 

50 

7 

785828 

50 

7 

790074 

49 

7 

794279 

49 

8 

781612 

58 

8 

785899 

57 

8 

790144 

56 

8 

794349 

56 

9 

781684 

65 

9 

785970 

64 

9 

790215 

63 

9 

794418 

63 

6050 

781755 

6110 

786041 

6170 

790285 

6230 

794488 

1 

781827   7 

1 

786112 

7 

1 

790356 

7 

1 

794558 

7 

2 

781899 

14 

2 

786183 

14 

2 

790426 

14 

'2 

794627 

14 

3 

781971 

22 

3 

786254 

21 

3 

790496 

21 

3 

794697 

21 

4 

782042 

29 

4 

786325 

28 

4 

790567 

28 

4 

794767 

28 

5 

782114 

36 

5 

786396 

36 

5 

790637 

35 

5 

794836 

35 

6 

782186 

43 

6 

786467 

43 

6 

790707 

42 

6 

794906 

42 

7 

782258 

50 

7 

786538 

50 

7 

790778 

49 

7 

794976 

49 

8 

782329 

58 

8 

786609 

57 

8 

790848 

56 

8 

795045 

56 

9 

782401 

65 

9 

786680 

64 

9 

790918 

63 

9 

795115 

63 

6060 

782473 

6120 

786751 

6180 

790988 

6240 

795185 

1 

782544 

7 

1 

786822 

7 

1 

791059 

7 

1 

795254 

7 

2 

782616 

14 

2 

786893|  14 

2 

791129 

14 

2 

795324 

14 

3 

782688 

21 

3 

786964 

21 

3 

791199 

21 

3 

795393 

21 

4 

782759 

29 

4 

787035 

28 

4 

791269 

28 

4 

795463 

28 

5 

782831 

36 

5 

787106 

36 

5 

791340 

35 

5 

795532 

35 

6 

782902 

43 

6 

787177 

43 

6 

791410 

42 

6 

795602 

42 

7 

782974 

50 

7 

787248 

50 

7 

791480 

49 

7 

795671 

49 

8 

783046 

57 

8 

787319 

67 

8 

791550 

56 

8 

795741 

56 

9 

783117 

64 

9 

787390 

64 

*9 

791620 

63 

9 

795810 

63 

6070 

783189 

6130 

787460 

6190 

791691 

6250 

795880 

1 

783260 

7 

1 

787531 

7 

1 

791761 

7 

1 

795949 

7 

2 

783332 

14 

2 

787602 

14 

2 

791831 

14 

2 

796019 

14 

3 

783403 

21 

3 

787673  21 

3 

791901 

21 

3 

796088 

21 

4 

783475 

29 

4 

787744  28 

4 

791971 

28 

4 

796158 

28 

5 

783546 

36 

5 

787815 

35 

5 

792041 

35 

5 

796227 

35 

6 

783618 

43 

6 

787885 

42 

6 

792111 

42 

6 

796297 

42 

7 

783689 

50 

7 

787956 

49 

7 

792181 

49 

-  7 

796366 

49 

8 

783761 

57 

8 

788027 

56 

8 

792252 

56 

8 

796436 

56 

9 

783832 

64 

9 

788098 

63 

9 

792322 

63 

9 

796505 

63 

6080 

783904 

6140 

788168 

6200 

792392 

6260 

796574 

1 

783975 

7 

1 

788239 

7 

1 

792462 

7 

1 

796644 

7 

2 

784046 

14 

2 

788310 

14 

2 

792532 

14 

2 

796713 

14 

3 

784118 

21 

3 

788381 

21 

3 

792602 

21 

3 

796782 

21 

4 

784189 

29 

4 

788451 

28 

4 

792672 

28 

4 

796852 

27 

5 

784261 

36 

5 

788522 

35 

5 

792742 

35 

5 

796921 

35 

6 

784332 

43 

6 

788593 

42 

6 

792812 

42 

6 

796990 

42 

7 

784403 

50 

7 

788663 

49 

7 

792882 

49 

7 

797060 

49 

8 

784475 

57 

8 

788734 

56 

8 

792952 

56 

8 

797129 

56 

9 

784546 

64 

9 

788804 

63 

9 

793022 

63 

9 

797198 

62 

6090 

784617 

6150 

788875 

6210 

793092 

6270 

797268 

1 

784689 

7 

1 

788946 

7 

1 

793162 

7 

1 

797337 

7 

2 

784760 

14 

2 

789016 

14 

2 

793231 

14 

2 

797406 

14 

3 

784831 

21 

3 

789087 

21 

3 

793301 

21 

3 

797475 

21 

4 

784902 

29 

4 

789157 

28 

4 

793371 

28 

4 

797545 

27 

5 

784974 

36 

5 

789228 

35 

5 

793441 

35 

5 

797614 

35 

6 

785045 

43 

6 

789299 

42 

6 

793511 

42 

6 

797683 

42 

7 

785116 

50 

7  1  789369 

49 

7 

793581 

49 

7 

797752 

49 

8 

785187 

57 

8 

789440 

56 

8 

793651 

56 

8 

797821 

56 

9 

785259 

64 

9 

789510 

63 

9 

793721 

63 

9 

797890 

62 

LOGARITHMS    OF    NUMBERS. 


25 


No. 

Log. 

S:  N°- 

Log. 

££  No- 

Log. 

Prop. 
Part. 

No. 

Log. 

i'ror* 
I'avt 

6280 

797960 

!;6340 

802089 

6400 

806180 

6460 

810233 

1 

798029 

7  ii   1 

802158 

7     1 

806248 

7 

1 

810300 

7 

2 

798098 

14  !    2 

802226 

14 

2 

806316 

14 

2 

810367 

13 

3 

798167 

21     3 

802295 

21 

3 

806384 

20 

3 

810434 

20 

4 

798236 

28     4 

802363 

27 

4 

806451 

27 

4 

810501 

27 

5 

798305 

34 

5 

802432 

34 

5 

806519 

34 

5 

810569 

33 

6 

798374 

41 

6 

802500 

41 

6 

806587 

41 

6 

810636 

40 

7 

798443 

48 

7 

802568 

48 

7 

806655 

48 

7 

810703 

47 

8 

798512 

55 

8 

802637 

55 

8 

806723 

54 

8 

810770 

54 

9 

798582 

62  ||   9 

802705 

62 

9 

806790 

61 

9 

810837 

60  i 

6290 

798651 

6350 

802774 

1  16410 

806858 

6470 

810904 

1 

798720 

7 

•1 

802842 

7 

1 

806926 

7 

1 

810971 

7 

2 

798789 

14 

2 

802910 

14 

2 

806994 

14 

2 

811038 

13 

3 

798858 

21 

3 

802979 

21 

3 

807061 

20 

3 

811106 

20 

4 

798927 

28 

4 

803047 

27 

4 

807129 

27 

4 

811173 

27 

0 

798996 

34 

5 

803116 

34 

5 

807197 

34 

5 

811240 

33 

6 

799065 

41 

6 

803184 

41 

6 

807264 

41 

6 

811307 

40 

7 

799134 

48 

7 

803252 

48 

7 

807332 

48 

7 

811374 

47 

8 

799203 

55 

8 

803320 

55 

8 

807400 

54 

8 

811441 

54 

9 

799272 

62 

9 

803389 

62 

9 

807467 

61 

9 

811508 

60 

6300 

799341 

16360 

803457 

6420 

807535 

6480 

811575 

1 

799409 

7 

1 

803525 

7 

1 

807603 

7 

1 

811642 

7 

2 

799478 

14 

2 

803594 

14 

2 

807670 

14 

2. 

811709 

13 

3 

799547 

21 

3 

803662 

21 

3 

807738 

20 

3 

811776 

20 

4 

799616 

28 

4 

803730 

27 

4 

807806 

27 

4 

811843 

27 

5 

799685 

34  !   5 

803798 

34 

5 

807873 

34 

5 

811910 

33  ! 

6 

799754 

41  1   6 

803867 

41 

6 

807941 

41 

6 

811977 

40  i 

7 

799823 

48  !   7 

803935 

48 

7 

808008 

48 

7 

812044 

47  I 

8 

799892 

55     8 

804003 

55 

8 

808076 

54 

8 

812111 

54 

9  799961 

62 

9 

804071 

62 

9 

808143 

61 

9 

812178 

60 

6310  800029 

6370 

804139 

6430 

808211 

6490 

812245 

| 

1  800098 

7 

1 

804208 

7 

1 

808279 

7 

1 

812312 

7 

2  !  800167 

14 

2 

804276 

14 

2 

808346 

14 

2 

812378 

13 

3  !  800236 

21 

3 

804344 

21 

3 

808414 

20 

3 

812445 

20 

4  1  800305 

28 

4 

804412 

27 

4 

808481 

27 

4 

812512 

27 

5  800373 

34 

5 

804480' 

34 

5 

808549 

.  34 

5 

812579 

33 

6  800442 

41 

6 

804548 

41 

6 

808616 

'  41 

6 

812646 

40 

7  800511 

48 

7 

804616 

48 

7 

808684 

48 

7 

812713 

47 

8  i  800580 

55 

8 

804685 

55 

8 

808751 

54 

8 

812780 

54 

9  800648 

62 

9 

804753 

62 

9 

808818 

61 

9 

812847 

60 

6320  800717 

6380 

804821 

6440 

808886 

6500 

812913 

1  800786 

7 

1 

804889 

7 

1 

808953 

7 

1 

812980 

7 

2  800854 

14 

2 

804957 

14 

2 

809021 

13 

2 

813047 

13 

3  800923 

21 

3 

805025 

20 

3 

809088 

20 

3 

813114 

20 

4 

800992 

28 

4 

805093 

27 

4 

809156 

27 

4 

813181 

27 

5 

801060 

34- 

5 

805161 

34 

5 

809223 

34 

5 

813247 

33 

6 

801129 

41 

6 

805229 

41 

6 

809290 

40 

6 

813314 

40 

7  |801198 

48 

7 

805297 

48 

7 

809358 

47 

7 

813381 

47 

8 

801266 

55 

8 

805365 

54 

8 

809425 

54 

8 

813448 

54 

9 

801335 

62 

9 

805433 

61 

9 

809492 

61 

9 

813514 

60 

6330  801404 

6390 

805501 

6450 

809560 

6510 

813581 

1 

801472 

7 

1 

805569 

7 

1 

809627 

7 

1 

813648 

7 

2 

801541 

14 

2 

805637 

14 

2 

809694 

13 

2 

813714 

13 

3 

801609 

21 

3 

805705 

20 

3 

809762 

20 

3 

813781 

20 

4 

801678 

27 

4 

805773 

27 

4 

809829 

27 

4 

813848 

27 

5 

801747 

34 

5 

805841 

34 

5 

809896 

34 

5 

813914 

33 

6 

801815 

41 

6 

805908 

41 

6 

809964 

40 

6 

813981 

40 

7 

801884 

48 

7 

805976 

48 

7 

810031 

47 

7 

814048 

47 

8 

801952 

55 

8 

806044 

54 

8 

810098 

54. 

8 

814114 

54 

9 

802021 

62 

9 

806112 

61     9 

810165 

61 

9 

814181 

60 

26 


LOGARITHMS    OF    NUMBERS. 


N, 

l-og. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Pan. 

No. 

Log. 

Prop. 
Part. 

No.  ,   Log. 

Prop. 
Part. 

6520 

814248 

6580 

818226 

6640 

822168 

6700  826075 

1 

814314 

7 

1 

818292 

7 

1 

822233 

7 

1  826140 

6 

2 

814381 

13 

-  2 

818358 

13 

2 

822299 

13 

2  ,  826204 

13 

3 

814447 

20 

3 

818424 

20 

3 

822364 

20 

3  j  826269 

19 

4 

814514 

26 

4 

818490 

26 

4 

822430 

26 

4  826334 

26 

5 

814581 

33 

5 

818556 

33 

5 

822495 

33 

5  826399  32 

6 

814647 

40 

6 

818622 

40 

6 

822560 

39 

6  |  826464  39 

7 

814714 

46 

7 

818688 

46 

7 

822626 

46 

7  !  826528 

45 

8 

814780 

53 

8 

818754 

53 

8 

822691 

52 

8  I  826593 

52 

9 

814847 

60 

0 

818819 

59 

9 

822756 

59 

9  826658 

58 

6530 

814913 

6590 

818885 

6650 

822823  i 

6710 

826722 

1 

814980 

7 

1 

818951 

7 

1 

822887 

7 

1 

826787 

6 

2 

815046 

13 

2 

819017 

-13 

2 

822952 

13 

2 

826852 

13 

3 

815113 

20 

0 

819083 

20 

3 

823018 

20 

3 

826917 

19 

4 

815179 

26 

4 

819149 

26 

4 

828088 

26 

4 

826981 

26 

5 

815246 

33 

5 

819215 

33 

5 

823148 

33 

5 

827046 

32 

6 

815312 

40 

6 

819281 

40 

6 

823213 

39 

6 

827111 

39 

7 

815378 

46 

7 

819346 

46 

7 

823279 

46 

7 

827175 

45 

8 

815445 

53 

8 

819412 

53 

8 

823344 

'52 

8 

827240 

52 

9 

815511 

60 

9 

819478 

59 

9 

823409 

59 

9 

827305 

58 

6540 

815578 

6600 

819544 

6660 

823474 

6720 

827369 

1 

815644 

7 

1 

819610 

7 

1 

823539 

7 

1 

827434 

6 

2 

815711 

13 

o 

41 

819675 

13 

2 

823005 

13 

2 

827498 

13 

3 

815777 

20 

3 

819741 

20 

3 

823070 

20 

3 

827563 

19 

4 

815843 

26 

4 

819807 

26 

4 

823735 

26 

4 

827628 

26 

5 

815910 

33 

5 

819873 

33 

5 

823800 

33 

5 

827692 

32 

6 

815976 

40 

6 

819939 

40 

6 

828866 

39 

6 

827757 

39 

7 

816042 

46 

7 

820004 

46 

7 

823930 

46 

7 

827821 

45 

8 

816109 

53 

8 

820070 

53 

8 

823996 

52 

8 

827886 

52 

9 

816175 

60 

9 

820136 

59 

9 

824061 

59 

9 

827951 

58 

6550 

816241 

6610 

820201 

6670 

824126 

6730 

828015 

1 

816308 

7 

1 

820267 

7 

1 

824191 

6 

1 

828080 

6 

2 

816374 

13 

2 

820333 

13 

2 

824256 

13 

2 

828144 

13 

3 

816440 

20 

3 

820399 

20 

3 

824321 

19 

3 

828209 

19 

4 

816506 

26 

4 

820464 

26 

4 

824386 

26 

4 

828273 

26 

5 

816573 

33 

6 

820530 

33 

5 

824451 

32 

5 

828338 

32 

6 

816639 

40 

6 

820595 

40 

6 

824516 

39 

6 

828402 

39 

7 

816705 

46 

7 

820661 

46 

7 

824581 

45 

7 

828467 

45 

8 

816771 

53 

8 

820727 

53 

8 

824646 

52 

8 

828531 

52 

9 

816838 

60 

9 

820792 

59 

9 

824711 

58 

9 

828595 

58 

6560 

816904 

6620 

820858 

6680 

824776 

6740 

828660 

1 

816970 

7 

1 

820924 

7 

1 

824841 

6 

1 

828724 

6 

2 

817036 

13 

2 

820989 

13 

2 

824906 

13 

2 

828789 

13 

3 

817102 

20 

3 

821055 

20 

3 

824971 

19 

3 

828853 

19 

4 

817169 

26 

4 

821120 

26 

4 

825036 

26 

4 

828918 

26 

5 

817235 

33 

5 

821186 

33 

5 

825101 

32 

5 

828982 

32 

6 

817301 

40 

6 

821251 

40 

6 

825166 

39 

6 

829046 

39 

7 

817367 

46 

7 

821317 

46 

7 

825231 

45 

7 

829111 

45 

8 

817433 

53 

8 

821382 

53 

8 

825296 

52 

8 

829175 

52 

9 

817499 

59 

9 

821448 

59 

9 

825361 

58 

9 

829239 

58 

6570 

817565 

6630 

821514 

6690 

825426 

6750 

829304 

1 

817631 

7 

1 

821579 

7 

1 

825491 

6 

1 

829368 

6 

2 

817698 

13 

2 

821644 

13 

2 

825556 

13 

2 

829432 

13 

3 

817764 

20 

3  821710 

20 

3 

825621 

19 

3 

829497 

19 

4 

817830 

26 

4  821775 

26 

4 

825686 

26 

4 

829561 

26 

5 

817896 

33 

5  821841  33 

5 

825751 

32 

5 

829625 

32 

6 

817962 

40 

6  821906  39 

6 

825815 

39 

6 

829690 

39 

7 

818028 

46 

7  821972  !  46 

7 

825880 

45 

7 

829754 

45 

8 

818094 

53 

8  822037 

52 

8 

825945 

52 

8 

8298181  62 

9 

818160 

59 

9  822103 

59  P   9 

826010 

58 

9 

829882|  58 

LOGARITHMS   OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

0,60 

829947 

6820 

833784 

6880 

837588 

6940 

841359 

1 

830011 

6 

1 

833848 

6 

1 

837652 

6 

1 

841422 

6 

2 

830075 

13 

2 

833912 

13 

2 

837715 

13 

2 

841485 

13 

3 

830139 

19 

3 

833975 

19 

3 

837778 

19 

3 

841547 

19 

1 

830204 

26 

4 

834039 

26 

4 

837841 

25 

4 

841610 

25 

5 

830268 

32 

5 

834103 

32 

5 

837904 

32 

5 

841672 

31 

6 

830332 

38 

6 

834166 

38 

6 

837967 

38 

6 

841735 

38 

7 

830396 

45 

7 

834230 

45 

7 

838030 

44 

7 

841797 

44 

8 

830460 

51 

8 

834293 

51 

8 

838093 

50 

8 

841860 

50 

9 

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58 

9 

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58 

9 

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57 

9 

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56 

6770 

830589 

6830 

834421 

6890 

838219 

6950 

841985 

1 

830653 

6 

1 

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6 

1 

838282 

6 

1 

842047 

6 

2 

830717 

13 

2 

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13 

2 

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13 

2 

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12 

3 

830781 

19 

3 

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19 

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19 

3 

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19 

4 

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26 

4 

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26 

4 

838471 

25 

4 

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25 

5 

830909 

32 

5 

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32 

5 

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32 

5 

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31 

6 

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38 

6 

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38 

6  838597 

38 

6 

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37 

7 

831037 

45 

7 

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45 

7 

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44 

7 

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44 

8 

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51 

8 

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51 

8 

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50 

8 

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50 

9 

831166 

58 

9 

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58 

9 

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57 

9 

842547 

56 

6780 

831230 

6840 

835056 

6900 

838849 

6960 

842609 

1 

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6 

1 

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6 

1 

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6 

1 

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6 

2 

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13 

2 

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18 

2 

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13 

2 

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12 

3 

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19 

3 

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19 

3 

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19 

3 

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19 

4 

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26 

4 

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26 

4 

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25 

4 

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25 

5 

831550 

32 

5 

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32 

5 

839164 

31 

5 

842921 

31 

6 

831614 

38 

6 

835437 

38 

6 

839227 

38 

6 

842983 

37 

7 

831678 

45 

7 

835500 

45 

7 

839289 

44 

7 

843046 

44 

8 

831742 

51 

8 

835564 

51 

8 

839352 

50 

8 

843108 

50 

9 

831806 

58 

9 

835627 

58 

9 

839415 

57 

9 

843170 

56 

6790 

831870 

6850 

835691 

6910 

839478 

6970 

843233 

1 

831934 

6 

1 

835754 

6 

1 

839541 

6 

1 

843295 

6 

2 

831998 

13 

2 

835817 

13 

2 

839604 

13 

2 

843357 

12 

3 

832062 

19 

3  835881 

19 

3 

839667 

19 

3 

843420 

19 

4 

832126 

26 

4 

835944 

26 

4 

839729 

25 

4 

843482 

25 

5 

832189 

32 

5 

836007 

32 

5 

839792 

31 

5 

843544 

31 

6 

832253 

38 

6 

836071 

38 

6 

839855 

38 

6 

843606 

37 

7 

832317 

45 

7 

836134 

45 

7 

839918 

44 

7 

843669 

43 

8 

832381 

51 

8 

836197 

51 

8 

839981 

50 

8 

843731 

50 

9 

832445 

58 

9 

836261 

58 

9 

840043 

57 

9 

843793 

56  i 

6800 

832509 

6860 

836324 

6920 

840106 

6980 

843855 

1 

832573 

6 

1 

836387 

6 

1 

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6 

1 

843918 

6 

2 

832637 

13 

2 

836451 

13 

2 

840232 

13 

2 

843980 

12 

3 

832700 

19 

3 

836514 

19 

3 

840294 

19 

3 

844042 

19 

4 

832764 

26 

4 

836577 

26 

4 

840357 

25 

4 

844104 

25 

5 

832828 

32 

5 

836641 

32 

5 

840420 

31 

5 

844166 

31 

6 

832892 

38 

6 

836704 

38 

6 

840482 

38 

6 

844229 

37 

7 

832956 

45 

7 

836767 

45 

7 

840545 

44 

7 

844291 

43 

8 

833020 

51 

8 

836830 

51 

8 

840608 

50 

8 

844353 

50 

9 

833083 

58 

9 

836894 

58 

9 

840671 

57 

9 

844415 

56 

6810 

833147 

6870 

836957 

6930 

840733 

6990 

844477 

- 

1 

833211 

6 

1 

837020 

6 

1 

840796 

6 

1 

844539 

6 

2 

833275 

13 

2 

837083 

13 

2 

840859 

13 

2 

844601 

12 

3 

833338 

19 

3 

837146 

19 

3 

840921 

19 

3 

844664 

19 

4 

833402 

26 

4 

831210 

25 

4 

840984 

25 

4 

844726 

25 

5 

833466 

32 

5 

837273 

32 

5 

841046 

31 

5 

844788 

31 

6 

833530 

38 

6 

837336 

38 

6 

841109 

38 

6 

844850 

37 

7 

833593 

45 

7 

837399 

44 

7 

841172 

44 

7 

844912 

43 

O 

833657 

51 

8 

837462 

51 

8 

841234 

50 

8  844974 

50 

9  833721 

58 

9 

837525 

57 

9 

841297 

56 

9  :  845036 

56 

28 


LOGARITHMS    OP   NUMBERS. 


No. 

Log. 

SE 

No.     Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

7000 

845098 

7060 

848805 

7120 

852480 

7180 

856124 

1 

845160 

6 

1 

848866 

6 

1 

852541 

6 

1 

856185 

6 

2 

845222 

12 

2 

848928 

12 

2 

852602 

12 

2 

856245 

12 

3 

845284 

19 

3 

848989 

18 

3 

852663 

18 

3 

856306 

18 

4 

845346 

25 

4  849051 

25 

4 

852724 

24 

4 

856366 

24 

5 

845408 

31 

5 

849112 

31 

5 

852785 

30 

5 

856427 

80 

6 

845470 

37 

6 

849174 

37 

6 

852846 

37 

6 

856487 

36 

7 

845532 

43 

7 

849235 

43 

7 

852907 

43 

7 

856548 

42 

8 

845594 

50 

8 

849296 

49 

8 

852968 

49 

8 

856608 

48 

9 

845656 

56 

9 

849358 

55 

9 

853029 

55 

9 

856668 

54 

7010 

845718 

7070 

849419 

7iao 

853090 

7190 

856729 

1 

845780 

6 

1 

849481 

6 

i 

853150 

6 

1 

856789 

6 

2 

845842 

12 

2 

849542 

12 

2 

853211 

12 

2 

856850 

12 

3 

845904 

19 

3 

849604 

18 

3 

853272 

18 

3 

856910 

18 

4 

845966 

25 

4 

849665 

25 

4 

853333 

24 

4 

856970 

24 

5 

846028 

31 

6 

849726 

31 

5 

853394 

30 

5 

857031 

30 

6 

846090 

37 

6 

849788 

37 

6 

853455 

37 

6 

857091 

36 

7 

846151 

43 

7 

849849 

43 

7 

853516 

43 

7 

857151 

42 

8 

846213 

50 

8 

849911 

49 

8 

853576 

49 

8 

857212 

48 

9 

846275 

56 

9 

849972 

55 

9 

853637 

55 

9 

857272 

54 

7020 

846337 

7080 

850033 

7140 

853698 

7200 

857332 

1 

846399 

6 

1 

850095 

6 

-  1 

853759 

6 

1 

857393 

6 

2 

846461 

12 

2 

850156 

12 

2 

853820 

12 

2 

857453 

12 

3 

846523 

19 

3 

850217 

18 

3 

853881 

18 

3 

857513 

18 

4 

846584 

25 

4 

850279 

25 

4 

853941 

24 

4 

857574 

24 

5 

846646 

31 

5 

850340 

31 

5 

854002 

30 

5 

857634 

30 

6 

846708 

37 

6 

850401 

37 

6 

854063 

37 

6 

857694 

36 

7 

846770 

43 

7 

850462 

43 

7 

854124 

43 

7 

857754 

42 

8 

846832 

50 

8 

850524 

49 

8 

854185 

49 

8 

857815 

48 

9 

846894 

56 

9 

850585 

55 

9 

854245 

55 

9 

857875 

54 

7030 

846955 

7090 

850646 

7150 

854306 

7210 

857935 

1 

847017 

6 

1 

850707 

6 

1 

854367 

6 

1 

857995 

6 

,  2 

847079 

12 

2 

850769 

12 

2 

854427 

12 

2 

858056 

12 

3 

847141 

19 

3 

850830 

18 

8 

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18 

3 

858116 

18 

4 

847202 

25 

4 

850891 

25 

4 

854549 

24 

4 

858176 

24 

5 

847264 

31 

5 

850952 

31 

5 

854610 

30 

5 

858236 

30 

6 

847326 

37 

6 

851014 

37 

6 

854670 

36 

6 

858297 

36 

7 

847388 

43 

7 

851075 

43 

7 

854731 

42 

7 

858357 

42 

8 

847449 

60 

8 

851136 

49 

8 

854792 

48 

8 

858417 

48 

9 

847511 

56 

9 

851197 

55 

9 

854852 

54 

9 

858477 

54 

7040 

847573 

7100 

851258 

7160 

854913 

7220 

858537 

1 

847634 

6 

1 

851320 

6 

1 

854974 

6 

1 

858597 

6 

2 

847696 

12 

2 

851381 

12 

2 

855034 

12, 

2 

858657 

12 

3 

847758 

18 

3 

851442 

18 

3 

855095 

18 

3 

858718 

18 

4 

847819 

25 

4 

851503 

25 

4 

855156 

24 

4 

858778 

24 

5 

847881 

31 

5 

851564 

31 

5 

855216 

30 

5 

858838 

30 

6 

847943 

37 

6 

851625 

37 

6 

855277 

36 

6 

858898 

36 

7 

848004 

43 

7 

851686 

43 

7 

855337 

42 

7 

858958 

42 

8 

848066 

49 

8 

851747 

49 

8 

855398 

48 

8 

859018 

48 

9 

848127 

55 

9 

851808 

55 

9 

855459 

54 

9 

859078 

54 

7050 

848189 

7110 

851870 

7170 

855519 

7230 

859138 

1 

848251 

6 

1 

851931 

6 

1 

855580 

6 

1 

859198 

6 

2 

848312 

12 

2 

851992 

12 

2 

855640 

12 

2 

859258 

12 

3 

848374 

18 

3 

852053 

18 

3 

855701 

18 

3 

859318 

18 

4 

848435 

25 

4 

852114 

25 

4 

855761 

24 

4 

859378 

24 

5 

848497 

31 

5 

852175 

31 

5 

855822 

30 

5 

859438 

30 

6 

848559 

37 

6 

852236 

37 

6 

855882 

36 

6 

859499 

36 

7 

848620 

43 

7 

852297 

43 

7 

855943 

42 

7 

859559 

42 

8 

848682 

49 

8 

852358 

49 

8 

856003 

48 

8 

859619 

48 

9 

848743 

55 

9 

852419 

55 

9 

856064 

54 

9 

859679 

54 

LOGARITHMS    OF   NUMBERS. 


29 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop.   -»j 
Part.;  No- 

Log. 

&t 

No. 

Log. 

S£ 

7240 

859739 

7300 

863323 

7360 

866878 

7420 

870404 

1 

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6 

1 

863382 

6 

1 

866937 

6 

1 

870462 

6 

2 

859858 

12 

2 

863442 

12 

2 

866996 

12 

2 

870521 

12 

3 

859918 

18 

3 

863501 

18 

3 

867055 

18 

3 

870579 

18 

4 

859978 

24 

4 

863561 

24 

4 

867114 

24 

4 

870638 

24 

5 

860038 

30 

5 

863620 

30 

5 

867173 

29 

5 

870696 

29 

6 

860098 

36 

6 

863680 

36 

6 

867232 

35 

6 

870755 

35 

7 

860158 

42 

7 

863739 

42 

7 

867291 

41 

7 

870813 

41 

8 

860218 

48 

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48 

8 

867350 

47 

8 

870872 

47 

9 

860278 

54 

9 

863858 

54 

9 

867409 

53 

9 

870930 

53 

7250 

860338 

7310 

863917 

7370 

867467 

7430 

870989 

1 

860398 

6 

1 

863977 

6 

1 

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6 

1 

871047 

6 

2 

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12 

2 

864036 

12 

2 

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12 

2 

871106 

12 

3 

860518 

18 

8 

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18 

3 

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18 

3 

871164 

18 

4 

860578 

24 

4 

864155 

24 

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867703 

24 

4 

871223 

24 

5 

860637 

30 

5 

864214 

30 

5 

867762 

29 

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871281 

29 

6 

860697 

36 

6 

864274 

36 

6 

867821 

35 

6 

871339 

35 

7 

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42 

7 

864333 

42 

7 

867880 

41 

7 

871398 

41 

8 

860817 

48 

8 

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48 

8 

867939 

47 

8 

871456 

47 

9 

860877 

54 

9 

864452 

54 

9 

867998 

53 

9 

871515 

53 

7260 

860937 

7320 

864511 

7380 

868056 

7440 

871573 

1 

860996 

6 

1 

864570 

6 

1 

868115 

6 

1 

871631 

6 

2 

861056 

12 

2 

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12 

2 

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12 

2 

871690 

12 

3 

861116 

18 

3 

864689 

18 

3 

868233 

18 

3 

871748 

18 

4 

861176 

24 

4 

864748 

24 

4 

868292 

24 

4 

871806 

23 

5 

861236 

30 

5 

864808 

30 

5 

868350 

29 

5 

871865 

29 

6 

861295 

36 

6 

864867 

36 

6 

868409 

35 

6 

871923 

35 

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861355 

42 

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864926 

42 

7 

868468 

41 

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41 

8  ;  861415 

48 

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48 

8 

868527 

47 

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872040 

47 

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54 

9 

865045 

54 

9 

868586 

53 

9 

872098 

53 

7270 

861534 

7330 

865104 

7390 

868644 

7450 

872156 

1 

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6 

1 

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6 

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6 

1 

872215 

6 

2 

861654 

12 

2 

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12 

2 

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12 

2 

872273 

12 

3 

861714 

18 

3 

865282 

18 

3 

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18 

3 

872331 

18 

4 

861773 

24 

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24 

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24 

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872389 

23 

5 

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30 

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29 

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29 

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36 

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36 

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35 

6 

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35 

7 

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42 

7 

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42 

7 

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41 

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41 

8 

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48 

8 

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48 

8 

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47 

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872622 

47 

9 

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54 

9 

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54 

9 

869173 

53  ! 

9 

872681 

53 

7280 

862131 

7340 

865696 

7400 

869232 

7460 

872739 

1 

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6 

1 

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6 

1 

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6  j 

1 

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2 

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12 

2 

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12 

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12 

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872855 

12 

3 

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18 

3 

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18 

3 

869408 

18 

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872913 

18 

4 

862370 

24 

4 

865933 

24 

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869466 

24 

4 

872972 

23 

5 

862430 

30 

5 

865992 

30 

5 

869525 

29 

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873030 

29 

6 

862489 

36 

6 

866051 

36 

6 

869584 

35 

6 

873088 

35 

7 

862549 

42 

7 

866110 

42 

7 

869642 

41 

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873146 

41 

8 

862608 

48 

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866169 

48 

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869701 

47 

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873204 

47 

9 

862668 

54 

9 

866228 

54 

9 

869760 

53  i 

9 

873262 

53 

7290 

862728 

7350 

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7410 

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7470 

873321 

1 

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6 

1 

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6 

1 

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2 

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12 

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12 

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12 

3 

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18 

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18 

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18  1 

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18 

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24 

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4 

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23 

5 

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30 

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36 

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35 

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35 

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42 

7 

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873727 

41 

8 

868204 

48 

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870287 

47 

8 

873785 

47 

9 

863263 

54 

9 

866819 

53 

9 

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53 

9 

873844 

53 

30 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 

Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No-     Log.    ££ 

7480 

873902 

7540 

877371 

7600 

880814 

7660 

884229 

1 

873960 

6 

1 

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6 

1 

880871 

6 

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12 

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11 

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11 

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874076 

17 

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17 

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17 

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17 

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23 

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23 

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23 

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884455 

23 

5 

874192 

29 

5 

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29 

5 

881099 

28 

5 

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28 

6 

874250 

35 

6 

877717 

34 

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34 

6 

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34 

7 

874308 

41 

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40 

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40 

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8 

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46 

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46 

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9 

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17 

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17 

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23 

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23 

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23 

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29 

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28 

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34 

6 

881727 

34 

6 

885135 

34 

7 

874887 

41 

7 

878349 

40 

7 

881784 

40 

7 

885192 

40 

8 

874945 

46 

8 

878407 

46 

8 

881841 

46 

8 

885248 

46 

9 

875003 

62 

9 

878464 

52 

9 

881898 

51 

9 

885305 

51 

7500 

875061 

7560 

878522 

7620 

881955 

7680 

885361 

1 

875119 

6 

1 

878579 

6 

1 

882012 

6 

1 

885418 

6 

2 

875177 

12 

2 

878637 

12 

2  882069 

11 

2 

885474 

11 

3 

875235 

17 

3 

878694 

17 

3  882126 

17 

3  !  885531 

17 

4 

875293 

23 

4 

878751 

23 

4  882183 

23 

4  885587 

23 

5 

875351 

29 

5 

878809 

29 

5  882240 

28 

5  885644 

28 

6 

875409 

35 

6 

878866 

34 

6 

882297 

34 

6  :  885700 

34 

7 

875466 

41 

7 

878924 

40 

7 

882354 

40 

7  ;  885757 

39 

8 

875524 

46 

8 

878981 

46 

8 

882411 

46 

8  i  885813 

45 

9 

875582 

52 

9 

879038 

62 

9 

882468 

51 

9  885870 

51 

7510 

875640 

7570 

879096 

7630 

882524 

7690 

885926 

1 

875698 

6 

1 

879153 

6 

1 

882581 

6 

1 

885983 

6 

2 

875756 

12 

2 

879211 

12 

2 

882638 

11 

2 

886039 

11 

3 

875813 

17 

3 

879268 

17 

3 

882695 

17 

3 

886096 

17 

4 

875871 

23 

4 

879325 

23 

4 

882752 

23 

4 

886152 

23 

*>  5 

875929 

29 

5 

879383 

29 

5 

882809 

28 

5 

886209 

28 

6 

875987 

35 

6 

879440 

34 

6 

882866 

34 

6 

886265 

34 

7 

876045 

41 

7 

879497 

40 

7 

882923 

40 

7 

886321 

39 

8 

876102 

46 

8 

879555 

46 

8 

882980 

46 

8 

886378 

45 

9 

876160 

52 

9 

879612 

52 

9 

883037 

51 

9 

886434 

51 

7520 

876218 

7580 

879669 

7640 

883093 

7700 

886491 

1 

876276 

6 

1 

879726 

6 

1 

883150 

6 

1 

886547 

6 

2 

876333 

12 

2 

879784 

11 

2 

883207 

11 

2 

886604 

11 

3 

876391 

17 

3 

879841 

17 

3 

883264 

17 

3 

886660 

17 

4 

876449 

23 

4 

879898 

23 

4 

883321 

23 

4 

886716 

23 

5 

876507 

29 

5 

879956 

28 

5 

883377 

28 

5 

886773 

28 

6 

876564 

34 

6 

880013 

34 

6 

883434 

34 

6 

886829 

34 

7 

876622 

40 

7 

880070 

40 

7 

883491 

40 

7 

886885 

39 

8 

876680 

46 

8 

880127 

46 

8 

883548 

46 

8 

886942 

45 

9 

876737 

52 

9 

880185 

51 

9 

883605 

51 

9 

886998 

51 

7530 

876795 

7590 

880242 

7650 

883661 

7710 

887054 

1 

876853 

6 

1 

880299 

6 

1 

883718 

6 

1 

887111 

6 

2 

876910 

12 

2 

880356 

11 

2 

883775 

11 

2 

887167 

11 

3 

876968 

17 

3 

880413 

17 

3 

883832 

17 

3 

887223 

17 

4 

877026 

23 

4 

880471 

23 

4 

883888 

23 

4 

887280 

23 

5 

877083 

29 

5 

880528 

28 

5 

883945 

28 

5 

887336 

28 

6 

877141 

34 

6  880585 

34 

6 

884002 

34 

6 

887392 

34 

7 

877198 

40 

7  1  880642 

40 

7 

884059 

40 

7 

887449 

39 

8 

877256 

46 

8  880699 

46 

8 

884115  46 

8 

887505 

45 

9 

877314 

52 

9  880756 

51 

9  884172!  51 

9 

887561 

51 

LOGARITHMS    OF    NUMBERS. 


31 


No. 

Log. 

Prop. 
Purt. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

7720 

887617 

7780 

890980 

7840 

894316 

7900 

897627 

1 

887674 

6 

1 

891035 

6 

1 

894371 

6 

1 

897682 

6 

2 

887730 

11 

2 

891091 

11 

2 

894427 

11 

2 

897737 

11 

3 

887786 

17 

3 

891147 

17 

3 

894482 

17 

3 

897792 

17  i 

4 

887842 

23 

4 

891203 

22 

4 

894538 

22 

4 

897847 

22 

5 

887898 

28 

5 

891259 

28 

5 

894593 

27 

5 

897902 

27 

6 

887955 

34 

6 

891314 

34 

6 

894648 

33 

6 

897957 

33 

7 

888011 

39 

7 

891370 

39 

7 

894704 

39 

7 

898012 

39 

8 

888067 

45 

8 

891426 

45 

8 

894759 

44 

8 

898067 

44 

9 

888123 

51 

9 

891482 

50 

9 

894814 

50 

9 

898122 

50 

7730 

888179 

7790 

891537 

7850 

894870 

7910 

898176 

1 

888236 

6 

1 

891593 

6 

1 

894925 

6 

1 

898231 

6 

2 

888292 

11 

2 

891649 

11 

2 

894980 

11 

2 

898286 

11 

3 

888348 

17 

3 

891705 

17 

3 

895036 

17 

3 

898341 

17 

4 

888404 

22 

4 

891760 

22 

4 

895091 

22 

4 

898396 

22 

5 

888460 

28 

5 

891816 

28 

5 

895146 

27 

5 

898451 

27 

6 

888516 

34 

6 

891872 

33 

6 

895201 

33 

6 

898506 

33 

7 

888573 

39 

7 

891928 

39 

7 

895257 

39 

7 

898561 

39 

8 

888629 

45 

8 

891983 

44 

8 

895312 

44 

8 

898615 

44 

9 

888685 

50 

9 

892039 

60 

9 

895367 

50 

9 

898670 

50 

7740 

888741 

7800 

892095 

7860 

895423 

7920 

898725 

1 

888797 

6 

1 

892150 

6 

1 

895478 

6 

1 

898780 

5 

2 

888853 

11 

2 

892206 

11 

2 

895533 

11 

2 

898835 

11 

3 

888909 

17 

3 

892262 

17 

3 

895588 

17 

3 

898890 

17 

4 

888965 

22 

4 

892317 

22 

4 

895643 

22 

4 

898944 

22 

5 

889021 

28 

5 

892373 

28 

5 

895699 

27 

5 

898999 

27 

6 

889077 

34 

6 

892429 

33 

6 

895754 

33 

6 

899054 

33 

7 

889134 

39 

7 

892484 

39 

7 

895809 

39 

7 

899109 

38 

8 

889190 

45 

8 

892540 

44 

8 

895864 

44 

8 

899164 

44 

9 

889246 

50 

a 

892595 

50 

9 

895920 

50 

9 

899218 

50 

7750 

889302 

7810 

892651 

7870 

895975 

7930 

899273 

1 

889358 

6 

1 

892707 

6 

1 

896030 

6 

1 

899328 

5 

2 

889414 

11 

2 

892762 

11 

2 

896085 

11 

2 

899383 

11 

3 

889470 

17 

3 

892818 

17 

3 

896140 

17 

3 

899437 

17 

4 

889526 

22 

4 

892873 

22     4 

896195 

22 

4 

899492 

22 

5 

889582 

28 

5 

892929 

28 

5 

896251 

27 

5 

899547 

27 

6 

889638 

34 

6 

892985 

33 

6 

896306 

33 

6 

899602 

33 

7 

889694 

39 

7 

893040 

39 

7 

896361 

39 

7 

899656 

38 

8 

889750 

45 

8 

893096 

44 

8 

896416 

44 

.8 

899711 

44 

9 

889806 

50 

9 

893151 

50 

9 

896471 

50 

9 

899766 

50 

7760 

889862 

7820 

893207 

7880 

896526 

7940 

899820 

1 

889918 

6 

1 

893262 

6 

1 

896581 

6 

1 

899875 

5 

2 

889974 

11 

2 

893318 

11 

2 

896636 

11 

2 

899930 

11 

3 

890030 

17 

3 

893373 

17 

3 

896692 

17 

3 

899985 

17 

4 

890086 

22 

4 

893429 

22 

4 

896747 

22 

4 

900039 

22 

5 

890141 

28 

5 

893484 

28     5 

896802 

27 

5 

900094 

27 

8 

890197 

34 

6 

893540 

33     6 

896857 

33 

6 

900149 

33 

7 

890253 

39 

7 

893595 

39  j   7 

896912 

39 

7 

900203 

38 

8 

890309 

45 

8 

893651 

44     8 

896967 

44 

8 

900258  44 

9 

890365 

50 

9 

893706 

50 

Q 

897022 

50 

9 

900312 

50 

7770 

890421 

7830 

893762 

7890 

897077 

7950 

900367 

1 

890477 

6 

1 

893817 

6 

1 

897132 

6 

1 

900422 

5 

2 

890533  11 

2 

893873 

11  I   2 

897187 

11 

2 

90047(5 

11 

3  890589 

17 

3 

893928 

17     3 

897242 

17 

3 

900531 

17 

4  |  890644 

22 

4 

893984 

22  1   4 

897297 

22 

4 

900586 

22 

5 

890700  28 
890756!  34 

5 
6 

894039 
894094 

28 
33 

5 

6 

897352 
897407 

27 
33 

5 

6 

900640 
900695 

27 
33  ' 

7 

890812 

39 

7 

894150 

39 

7 

897462 

39 

7 

900749 

38 

8 
9 

890868 
890924 

45 
50 

8 
9 

894205 
894261 

44 
50 

81897517 
9  897572 

44 
50 

8 
9 

900804 
900858 

44 
50 

32 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Pare. 

No. 

Log. 

Prop. 
Part. 

7960 

900913 

8020 

904174 

8080 

907411 

8140 

910624 

1 

900968 

5 

1 

904228 

5 

1 

907465 

5 

1 

910678 

5 

2 

901022 

11 

2 

904283 

11 

2 

907519 

11 

2 

910731 

11 

3 

901077 

16 

3 

904337 

16 

3 

907573 

16 

3 

910784 

16 

4 

901131 

22 

4 

904391 

22 

4 

907626 

22 

4 

910838 

21 

5 

901186 

27 

5 

904445 

27 

5 

907680 

27 

5 

910891 

27 

6 

901240 

33 

6 

904499 

32 

6 

907734 

32 

6 

910944 

32 

7 

901295 

38 

7 

904553 

38 

7 

907787 

38 

7 

910998 

37 

8 

901349 

44 

8 

904607 

43 

8 

907841 

43 

8 

911051 

43 

9 

901404 

49 

9 

904661 

49 

9 

907895 

49 

9 

911104 

48 

7970 

901458 

8030 

904715 

8090 

907948 

8150 

911158 

1 

901513 

5 

1 

904770 

5 

1 

908002 

5 

1 

911211 

5 

2 

901567 

11 

2 

904824 

11 

2 

908056 

11 

2 

911204 

11 

3 

901622 

16 

3 

904878 

16 

3 

908109 

16 

3 

911317 

16 

4 

901676 

22 

4 

904932 

22 

4 

908163 

22 

4 

911371 

21 

5 

901731 

27 

5 

904986 

27 

5 

908217 

27 

5 

911424 

27 

6 

901785 

33 

6 

905040 

32 

6 

908^70 

32 

6 

911477 

32 

7 

901840 

38 

7 

905094 

38 

7 

908324 

38 

7 

911530 

37 

8 

901894 

44 

8 

905148 

43 

8 

908378 

43 

8 

911584 

42 

9 

901948 

49 

9 

905202 

49 

9 

908431 

49 

9 

911637 

48 

7980 

902003 

8040 

905256 

8100 

908485 

8160 

911690 

1 

902057 

5 

1 

905310 

5 

1 

908539 

5 

1 

911743 

5 

2 

902112 

11 

2 

905364 

11 

2 

908592 

11 

2 

911797 

11 

3 

902166 

16 

3 

905418 

16 

3 

908646 

16 

3 

911850 

16 

4 

902221 

22 

4 

905472 

22 

4 

908699 

21 

4 

911903 

21 

5 

902275 

27 

5 

905526 

27 

5 

908753 

27 

5 

911956 

27 

6 

902329 

33 

6 

905580 

32 

6 

908807 

32 

6 

912009 

32 

7 

902384 

38 

7 

905634 

38 

7 

908860 

37 

7 

912063 

37 

8 

902438 

44 

8 

905688 

43 

8 

908914 

43 

8 

912116 

42 

9 

902492 

49 

9 

905742 

49 

9 

908967 

48 

9 

912169 

48 

7990 

902547 

8050 

905796 

8110 

909021 

8170 

912222 

1 

902601 

5 

1 

905850 

5 

1 

909074 

5 

1 

912275 

5 

2 

902655 

11 

2 

905904 

11 

2 

909128 

11 

2 

912328 

11 

3 

902710 

16 

3 

005958 

16 

3 

909181 

16 

3 

912381 

16 

4 

902764 

22 

4 

906012 

22 

4 

909235 

21 

4 

912435 

21 

5 

902818 

27 

5 

906065 

27 

5 

909288 

27 

5 

912488 

27 

6 

902873 

33 

6 

906119 

32 

6 

909342 

32 

6 

912541 

32 

7 

902927 

38 

7 

906173 

38 

7 

909395 

37 

7 

912594 

37 

8 

902981 

44 

8 

906227 

43 

8 

909449 

43 

8 

912647 

42 

9 

903036 

49 

9 

906281 

49 

9 

909502 

48 

9 

912700 

48 

3000 

903090 

8060 

906335 

8120 

909556 

8180 

912753 

1 

903144 

5 

1 

906389 

5 

1 

909609 

5 

1 

912806 

5 

2 

903198 

11 

2 

906443 

11 

2 

909663 

11 

2 

912859 

11 

3 

903253 

16 

3 

906497 

16 

3 

909716 

16 

3 

912913 

16 

4 

903307 

22 

4 

906550 

22 

4 

909770 

21 

4 

912966 

21 

5 

903361 

27 

5 

906604 

27 

5 

909823 

27 

5 

913019 

27 

6 

903416 

32 

6 

906658 

32 

6 

909877 

32 

6 

913072 

32 

7 

903470 

38 

7 

906712 

38 

7 

909930 

37 

7 

913125 

37 

8 

903524 

43 

8 

906766 

43 

8 

909984 

43 

8 

913178 

42 

9 

903578 

49 

9 

906820 

49 

9 

910037 

48 

9 

913231 

48 

8010 

903632 

8070 

906873 

8130 

910090 

8190 

913284 

1 

903687 

5 

1 

906927 

5 

1 

910144 

5 

1 

913337 

5 

2 

903741 

11 

2 

906981 

11 

2 

910197 

11 

2 

913390 

11 

3 

903795 

16 

3 

907035 

16 

3 

910251 

16 

3 

913443 

16 

4 

903849 

22 

4 

907089 

22 

4 

910304 

21 

4 

913496 

21 

5 

903903 

27 

5 

907142 

27 

5 

910358 

27 

5 

913549 

27 

6 

903958 

32 

6 

907196 

32 

6 

910411 

32 

6 

913602 

32 

7 

904012 

38 

7 

907250 

38 

7 

910464 

37 

7 

913655 

37 

8 

904066 

43 

8 

907304 

43 

8 

910518 

43 

8 

913708 

42 

9 

904120 

49 

9 

907358 

49 

9 

910571 

48 

9 

913761 

48 

LOGARITHMS    OF 


*"  CAUFQ^         33 


No. 

Log. 

JLT 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

8200 

913814 

8260 

916980 

8320 

920123 

8380 

923244 

1 

913867 

5 

1 

917033 

5 

1 

920175 

5 

1 

923296 

5 

2 

913920 

11 

2 

917085 

11 

2 

920228 

10 

2 

923348 

10 

3 

9*3973 

16 

3 

917138 

16 

3 

920280 

16 

3 

923399 

16 

4 

914026 

21 

4 

917190 

21 

4 

920332 

21 

4  923451 

21 

5 

914079 

27 

5 

917243 

26 

5 

920384 

26 

5  1  923503 

26 

6 

914131 

32 

6 

917295 

31 

6 

920436 

31 

6 

923555 

31 

7 

914184 

37 

7 

917348 

37 

7 

920489 

36 

7 

923607 

36 

8 

914237 

42 

8 

917400 

42 

8 

920541 

42 

8 

923658 

42 

9 

914290 

48 

9 

917453 

47 

9 

920593 

47 

9 

923710 

47 

8210 

914343 

8270 

917505 

8330 

920645 

8390 

923762 

1 

914396 

5 

1 

917558 

5 

1 

920697 

5 

1 

923814 

5 

2 

914449 

11 

2 

917610 

11 

2 

920749 

10 

2 

923865 

10 

3 

914502 

16 

3 

917663 

16 

3 

920801 

16 

3 

923917 

16 

4 

914555 

21 

4 

917715 

21 

4 

920853 

21 

4 

923969 

21 

5 

914608 

27 

5 

917768 

26 

5 

920906 

26 

5 

924021 

26 

6 

914660 

32 

6 

917820 

31 

6 

920958 

31 

6 

924072 

31 

7 

914713 

37 

7 

917873 

37 

7 

921010 

36 

7 

924124 

36 

8 

914766 

42 

8 

917925 

42 

8 

921062 

42 

8 

924176 

42 

9 

914819 

48 

9 

917978 

47 

9 

921114 

47 

9 

924228 

47 

8220 

914872 

8280 

918030 

8340 

921166 

8400 

924279 

1 

914925 

5 

1 

918U83 

5 

1 

921218 

5 

1 

924331 

5 

2 

914977 

11 

o 

918135 

11 

2 

921270 

10 

2 

924383 

10 

3 

915030 

16 

3 

918188 

16 

3 

921322 

16 

3 

924434 

15 

4 

915083 

21 

4 

918240 

21 

4 

921374 

21 

4 

924486 

21 

5 

915136 

27 

5 

918292 

26 

5 

921426 

26 

5 

924538 

26 

6 

915189 

32 

6 

918345 

31 

6 

921478 

31 

6 

924589 

31 

7 

915241 

37 

7 

918397 

37 

7 

921530 

36 

7 

924641 

36 

8 

915294 

42 

8 

918450 

42 

8 

921582 

42 

8 

924693 

41 

9 

915347 

48 

9 

918502 

47 

9 

921634 

47 

9 

924744 

46 

8230 

915400 

8290 

918555 

8350 

921686 

8410 

924796 

1 

915453 

5 

1 

918607 

5 

1 

921738 

5 

1 

924848 

5 

2 

915505 

11 

2 

918659 

11 

2 

921790 

10 

2 

924899 

10 

3 

915558 

16 

8 

918712 

16 

3 

921842 

16 

3 

924951 

15 

4 

915611 

21 

4 

918764 

21 

4 

921894 

21 

4 

925002 

21 

5 

915664 

27 

5 

918816 

26 

5 

921946 

26 

5 

925054 

26 

6 

915716 

32 

6 

918869 

31 

6 

921998 

31 

6 

925106 

31 

7 

915769 

37 

7 

918921 

37 

7 

922050 

36 

7 

925157 

36 

8 

915822 

42 

8 

918973 

42 

8 

922102 

42 

8 

925209 

41 

9 

915874 

48 

9 

919026 

47 

9 

922154 

47 

9 

925260 

46 

8240 

915927 

8300 

919078 

8360 

922206 

8420 

925312 

1 

915980 

5 

1 

919130 

5 

1 

922258 

5 

1 

925364 

5 

2 

916033 

11 

2 

919183 

11 

2 

922310 

10 

2 

925415 

10 

3 

916085 

16 

3 

919235 

16 

3 

922362 

16 

3 

925467 

15 

4 

916138 

21 

4 

919287 

21 

4 

922414 

21 

4 

925518 

21 

5 

916191 

27 

5 

919340 

26 

5 

922466 

26 

5 

925570 

26 

6 

916243 

32 

6 

919392 

31 

6 

922518 

31 

6 

925621 

31 

7 

916296 

37 

7 

919444 

37 

7 

922570 

36 

7 

925673 

36 

8 

916349 

42 

8 

919496 

42 

8 

922622 

42 

8 

925724 

41 

9 

916401 

48 

9 

919549 

47 

9 

922674 

47 

9 

925776 

46 

8250 

916454 

8310 

919601 

8370 

922725 

8430 

925828 

1 

916507 

5 

1 

919653 

5 

1 

922777 

6 

1 

925879 

5 

2 

916559 

11 

2 

919705 

11 

2 

922829 

10 

2 

925931 

10 

3 

916612 

16 

3 

919758 

16 

3 

922881 

16 

3 

925982 

15 

4 

916664 

21 

4 

919810 

21 

4 

922933 

21 

4 

926034 

21 

5 

916717 

26 

5 

919862 

26 

5 

922985 

26 

5 

926085 

26 

6 

916770 

31 

6 

919914 

31 

6 

923037 

31 

6 

926137 

31 

7 

916822 

37 

7  919967 

37 

7 

923088 

36 

7 

926188 

36 

8 

916875 

42 

8  920019 

42 

8 

923140 

42 

8 

926239 

41 

9 

916927 

47 

9  920071 

47  1   9  1  923192 

47 

9 

926291 

46 

34 


LOGARITHMS    OF   NUMBERS. 


No.   [   Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop 
Part. 

No. 

Log. 

Prop. 
Part, 

8440  '.  926342 

8500 

929419 

8560 

932474 

8620 

935507 

1  i  926394 

5 

1 

929470 

5 

1 

932524 

5 

1 

985558 

5 

2  926445 

10 

2 

929521 

10 

2 

932575 

10 

2 

935608 

10 

3  926497 

15 

3 

929572 

15 

3 

932626 

15 

3 

935,658 

15 

4  |  926548 

21 

4 

929623 

20 

4 

932677 

20 

4 

935709 

20 

5  926600 

26 

5 

929674 

26 

5 

932727 

25 

5 

935759 

25 

6  926651 

31 

6 

929725 

31 

6 

932778 

30 

6 

935809 

30 

7  !  926702 

36 

7 

929776 

36 

7 

932829 

35 

7 

935860 

35 

8  926754 

41 

8 

929827 

41 

8 

932879 

40 

8 

935910 

40 

9 

926805 

46 

9 

929878 

46 

9 

932930 

45 

9 

935960 

45 

8450  926857 

8510 

929930 

8570 

932981 

8680 

936011 

1 

926908 

5 

1 

929981 

5 

1 

933031 

5 

1 

936061 

5 

2 

926959 

10 

2 

930032 

10 

2 

933082 

10 

n 

93G111 

10 

3 

927011 

15 

3 

930083 

15    3 

933133 

15 

8 

936162 

15 

4 

927062 

21 

4 

930134 

20     4 

933183 

20 

4 

936212 

20 

5 

927114 

26 

5 

930185 

26  !   5 

933234 

25 

5 

936262 

25 

6 

927165 

31 

6 

930236 

31     6 

933285 

30 

6 

936318 

30 

7 

927216 

36 

7 

930287 

36     7 

933335 

35 

7 

936363 

35 

8 

927268 

41 

8 

930338 

41  !i   8 

933386 

40 

8 

936413 

40 

9 

927319 

46 

9 

930389 

46  ||   9 

933437 

45 

9 

936463 

45 

8460 

927370 

8520 

930440 

1  8580 

933487 

8640 

936514 

1 

927422 

5 

1 

930491 

5  ;      1 

933538 

5 

1 

936564 

5 

2 

927473 

10 

2 

930541 

10  \l   2 

933588 

10 

2 

936614 

10 

3 

927524 

15 

3 

930592 

15  *   3 

933639 

15 

3 

936664 

15 

4 

927576 

21 

4 

930643 

20     4 

933690 

20 

4 

936715 

20 

5 

927627 

26 

5 

930694 

25     5 

933740 

25 

5 

936765 

25 

6 

927678 

31 

6 

930745 

31  ii   6 

933791 

30 

6 

936815 

30 

7 

927730 

36 

7 

930796 

36  i    7 

933841 

35 

7 

936865 

35 

8 

927781 

41 

8 

930847 

41  i   8 

933892 

40 

8 

936916 

40 

9 

927832 

46 

9 

930898 

46  j|   9 

9339,43 

45 

9  936966 

45 

8470 

927883 

8530 

930949 

I  8590  j  933993 

8650  937016 

1 

927935 

5 

1 

931000 

5  •!   1  934044 

5 

1  937066 

5 

2 

927986 

10 

2 

931051 

10  1   2  934094 

10 

2  937116 

10 

3 

928037 

15 

3 

931102 

15     3  934145 

15 

3  937167 

15 

4 

928088 

21 

4 

931153 

20  !   4  934195 

20 

4 

937217 

20 

5 

928140 

26 

5 

931203 

25     5  934246 

25 

5 

937267 

25 

6 

928191 

31 

6 

931254 

81  ll   6  934296 

30 

6 

937317 

30 

7 

928242 

36 

7 

931305 

36     7  934347 

35 

7 

937367 

35 

8 

928293 

41 

8 

931356 

41     8  934397 

40 

8 

937418 

40 

9 

928345 

46 

9 

931407 

46 

9  |  934448 

45 

9 

937468 

45 

8480 

928396 

8540 

931458 

8600  934498 

8660 

937518 

1 

928447 

5 

1 

931509 

5 

1  934549 

5 

1 

937568 

5 

2 

928498 

10 

2 

931560 

10 

2  934599 

10 

2 

937618 

10 

3 

928549 

15 

3 

931610 

15 

3 

934650 

15 

3 

937668 

15 

4 

928601 

21 

4 

931661 

20 

4 

934700 

20 

4 

937718 

20  i 

5 

928652 

26 

5 

931712 

25 

5 

934751 

25 

5  937769 

25 

6 

928703 

31 

6 

931763 

31 

6 

934801 

30 

6  937819 

30 

7 

928754 

36 

7 

931814 

36 

7 

934852 

35 

7 

937869 

35 

8 

928805 

41 

8 

931864 

41 

8 

934902 

40 

8 

937919 

40 

9 

928856 

46 

9 

931915 

46 

9 

934953 

45 

9 

937969 

45 

8490 

928908 

8550 

931966 

8610 

935003 

8670 

938019 

1 

928959 

,  5 

1 

932017 

5 

1  935054 

5 

1 

938069 

5 

2 

929010 

10 

2 

932068 

10 

2  935104 

10 

2 

938119 

10 

3 

929061 

15 

3 

932118 

15 

3  935154 

15 

3 

938169 

15 

4 

929112 

20 

4 

932169 

20 

4 

935205 

20 

4 

938219 

20 

5 

929163 

26 

5 

932220 

25 

5 

935255 

25 

5 

938269 

25 

6 

929214 

31 

6 

932271 

30 

6 

935306  I  30 

6 

938319 

30 

7 

929266 

36 

7 

932321 

35 

7 

935356 

35 

7 

938370 

35 

8 

929317 

41 

8 

932372 

40 

8 

935406 

40 

8  938420 

40 

9 

929368 

46 

9 

932423 

45 

9 

935457 

45 

9  938470  45 

LOGARITHMS    OF    NUMBERS. 


35 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

££.  »'•    L'*- 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

8680 

938520 

8740 

941511 

8800 

944483 

8860 

947434 

1 

938570 

5 

1 

941561 

5 

1 

944532 

5  1    1 

947483 

5 

2 

938620 

10 

2 

941611 

10  i 

2 

944581 

10    2 

947532 

10 

3 

938670 

15 

3 

941660 

15 

3 

944631 

15 

3 

947581 

15 

4 

938720 

20 

4 

941710 

20 

4 

944680 

20 

4 

947630 

20 

5 

938770 

25 

5 

941760 

25 

5 

944729 

25 

5 

947679 

25 

6 

938820 

30 

6 

941809 

30 

6 

944779 

30 

6 

947728 

29 

7 

938870 

35 

7 

941859 

35 

7 

944828 

35 

7 

947777 

34 

8 

938920 

40 

8 

941909 

40 

8 

944877 

40 

8 

947826 

39  | 

9 

938970 

45 

9 

941958 

45 

9 

944927 

45 

9 

947875 

44 

8690 

939020 

8750 

942008 

8810 

944976 

8870 

947924 

1 

939070 

5 

1 

942058 

5 

1 

945025 

5     1 

947973 

5 

2 

939120 

10 

2 

942107 

10 

2 

945074 

10    2 

948021 

10 

3 

939170 

15 

3 

942157 

15 

3 

945124 

15  !   3 

948070 

15 

4 

939220 

20 

4 

942206 

20 

4 

945173 

20  i   4 

948119 

20 

5 

939270 

25 

5 

942256 

25 

5 

945222 

25    5 

948168 

25 

6 

939319 

30 

6 

942306 

30 

6 

945272 

30     6 

948217 

29 

7 

939369 

35 

7 

942355 

35 

7 

945321 

35     7 

948266 

34 

8 

939419 

40 

8 

942405 

40 

8 

945370 

40     8 

948315  39  | 

9 

939469 

45 

9 

942454 

45 

9 

945419 

45    9 

948364  44 

8700 

939519 

8760 

942504 

8820 

945469 

8880 

948413 

1 

939569 

5 

1 

942554 

5 

1 

945518 

5     1 

'948462 

5 

2 

939619 

10 

2 

942603 

10 

2 

945567 

10    2 

948511 

10 

3 

939669 

15 

3 

942653 

15 

3 

945616 

15  i   3 

9485601  15 

4 

939719 

20 

4 

942702 

20 

4 

945665 

20     4 

948608  i  20 

5 

939769 

25 

5 

942752 

25 

5 

945715 

25     5 

948657 

25 

6 

939819 

30 

6 

942801 

30 

6 

945764 

29 

6 

948706 

29 

7 

939868 

35 

7 

942851 

35 

7 

945813 

34 

7 

948755!  34 

8  939918 

40 

8 

942900 

40 

8 

945862 

39 

8 

948804 

39 

9  939968 

45 

9 

942950 

45 

9 

945911 

44  II   9 

948853 

44 

8710  940018 

8770 

943000 

8830 

945961 

8890 

948902 

1  940068 

5 

1 

943049 

5 

1 

946010 

5  II   1 

948951 

5 

2  i  940118 

10 

2 

943099 

10 

2 

946059 

10  !    2 

948999 

10- 

3 

940168 

15 

3 

943148 

15 

3 

946108 

15     3 

949048 

15 

4 

940218 

20 

4 

943198 

20 

4 

946157 

20     4 

949097 

20 

5 

940267 

25 

5 

943247 

25 

5 

946207 

25  |i   5 

949146 

25 

6 

940317 

30 

6 

943297 

30 

6 

946256 

29 

6 

949195 

29 

7 

940367. 

35 

7 

943346 

35 

7 

946305 

34 

7 

949244 

34 

8 

940417 

40 

8 

943396 

40 

8 

946354 

39 

8 

949292 

39 

9 

940467 

45 

9 

943445 

46 

9 

946403 

44  i 

9 

949341 

44 

8720 

940516 

8780 

943494 

8840 

946452 

1 

8900 

949390 

1 

940566 

5 

1 

943544 

5 

1 

946501 

'  5 

1 

949489 

5 

2  940616 

10 

2 

943593 

10 

2 

946550 

10  i!   2 

949488 

10 

3  940666 

15 

3 

943643 

15 

3 

946600 

15 

3 

949536 

15 

4  940716 

20 

4 

943692 

20 

4 

946649 

20 

4 

949585 

20 

5  940765 

25 

5 

943742 

25 

5 

946698 

25 

5 

949634 

25 

6  940815 

30 

6 

943791 

30 

6 

946747 

29 

6 

949683 

29 

7  i  940865 

35 

7 

943841 

35 

7 

946796 

34 

7 

949731 

34 

8  940915 

40 

8 

943890 

A 

8 

946845 

39 

8 

949780 

39 

9  ;  940964 

45 

9 

943939 

45 

9 

946894 

44 

9 

949829 

44 

8730  941014 

8790 

943989 

8850 

946943 

8910 

949878 

1 

941064 

5 

1 

944038 

5 

1 

946992 

5 

1 

949926 

5 

2  941114 

10 

2 

944088 

10 

2 

947041 

10 

2  949975 

10 

31941163 

15 

3 

944137 

15 

3 

947090 

15 

3  950024 

15 

4  941213 

20 

4 

944186 

20 

4 

947139 

20 

4  950073 

20 

5  941263 

25 

5 

944236 

25 

5 

947189 

25 

5  950121 

25 

6  941313 

30 

6 

944285 

30 

6 

947238 

29 

6  950170 

29 

7  941362  35 

7  944335 

35 

7 

947287 

34 

7  950219 

34 

8  941412  40 

8  I  944384 

40 

8 

947336 

39 

8  950267 

39 

9  941462  45 

9  !  944433 

45 

9 

947385 

44 

9  950316 

44 

34 


36 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No.     Log. 

Prop. 
Part. 

No. 

i*- 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

8920 

950366 

8980  j  953276 

9040 

956168 

9100 

959041 

1 

950413 

5 

1  |  953325 

5 

1 

956216 

5 

1 

959089 

5 

2 

950462 

10 

2  953373 

10 

2 

956264 

10 

2 

959137 

10 

3 

950511 

15 

3  953421 

15 

3 

956312 

14 

3 

959184 

14 

4 

950560 

19 

4  953470 

19 

4 

956361 

19 

4 

959232 

19 

5 

950608 

24 

5 

953518 

24 

5 

956409 

24 

5 

959280 

24 

6 

950657 

29 

6 

953566 

29 

6 

.956457 

29 

6 

959328 

29 

7 

950705 

34 

7 

953615 

34 

7 

956505 

34 

7 

959375 

34 

8 

950754 

39 

8 

953663 

39 

8 

956553 

38 

8 

959423 

38 

9 

950803 

44 

9 

953711 

44 

9 

956601 

43 

9 

959471 

43 

8930 

950851 

8990 

953760 

9050 

956649 

9110 

959518 

1 

950900 

5 

1 

953808 

6 

1 

956697 

5 

1 

959566 

5 

2 

950949 

10 

2 

953856 

10 

2 

956745 

10 

2 

959614 

10 

3 

950997 

15 

3 

953905 

15 

3 

956792 

14 

3 

959661 

14 

4 

951046 

19 

4 

953953 

19 

4 

956840 

19 

4 

959709 

19 

5 

951095 

24 

5 

954001 

24 

5 

956888 

24 

5 

959757 

24 

6 

951143 

29 

6 

954049 

29 

6 

956936 

29 

6 

959804 

29 

7 

951192 

34 

7 

954098 

34 

7 

956984 

34 

7 

959832 

34 

8 

951240 

39 

8 

954146 

39 

8 

957032 

38 

8 

959900 

38 

9 

951289 

44 

9 

954194 

44 

9 

957080 

43 

9 

959947 

43 

8940 

951337 

9000 

954242 

9060 

957128 

9120 

959995 

1 

951386 

5 

1 

954291 

5 

1 

957176 

5 

1 

960042 

5 

2 

951435 

10 

2 

954339 

10 

2 

957224 

10 

2 

960090 

10 

3 

951483 

15 

3 

954387 

14 

3 

957272 

14 

3 

960138 

14 

4 

951532 

19 

4 

954435 

19 

4 

957320 

19 

4 

960185 

19 

6 

951580 

24 

5 

954484 

24 

5 

957368 

24 

5 

960233 

24 

6 

951629 

29 

6 

954532 

29 

6 

957416 

29 

6 

960280 

28 

•   7 

951677 

34 

7 

954580 

34 

7 

957464 

34 

7 

960328 

33 

8 

951726 

39 

8 

954628 

38 

8 

957511 

38 

8 

960376 

38 

9 

951774 

44 

9 

954677 

43 

9 

957559 

43 

9 

960423 

43 

8950 

951823 

9010 

954725 

9070 

957607 

9130 

960471 

1 

951872 

5 

1 

954773 

5 

1 

957655 

5 

1 

960518 

5  : 

2 

951920 

10 

2 

954821 

10 

2 

957703 

10 

2 

960566 

10 

3 

951969 

15 

3 

954869 

14 

3 

957751 

14 

3 

960613 

14 

4 

952017 

19 

4 

954918 

19 

4 

957799 

19 

4 

960661 

19 

6 

952066 

24 

5 

9549661  24 

5 

957847 

24 

5 

960709 

24 

8 

952114 

29 

6 

955014 

29 

6 

957894 

29 

6 

960756 

28 

7 

952163 

34 

7 

955062 

34 

7 

957942 

34 

7 

960804 

33 

8 

952211 

39 

8 

955110 

38 

8 

957990 

38 

8 

960851 

38 

9 

952259 

44 

9 

955158 

43 

9 

958038 

43 

9 

960899 

43 

8960 

952308 

9020 

955206 

9080 

958086 

9140 

960946 

1 

,952356. 

5 

"  1 

955255 

5 

1 

958134 

5 

1 

960994 

5 

2 

952405 

10 

2 

955303 

10 

2 

958181 

10 

2 

961041 

10 

3 

952453 

15 

3 

955351 

14 

3 

958229 

14 

3 

961089 

14 

4 

952502 

19 

4 

955399 

19 

4 

958277 

19 

4 

961136 

19 

5 

952550 

24 

5 

955447 

24 

5 

958325 

24 

5 

961184 

24 

6 

952599 

29 

6 

955495 

29 

6 

958373 

29 

6 

961231 

28 

7 

952647 

34 

7 

955543 

34 

7 

958420 

34 

7 

961279 

33 

8 

952696 

39 

8 

955592 

38 

'   8 

958468 

38 

8 

96132(5 

38 

9 

952744 

44 

9 

955640 

43 

9 

958516 

43 

9 

961374 

43 

8970 

952792 

9030 

955688 

9090 

958564 

9150 

961421 

1 

952841 

5 

1 

955736 

5 

1 

958612 

5 

1 

961469 

5 

2 

952889 

10 

2 

955784 

10 

2 

958659 

10 

2 

961516 

10 

3 

952938 

15 

3 

955832 

14 

3 

958707 

14 

3 

961563 

14 

4 

952986 

19 

4 

955880 

19 

4 

958755 

19  1 

4 

961611 

19 

5 

953034 

24 

5 

955928 

24 

5 

958803 

24 

5 

961658 

24 

6 

953083 

29 

6 

955976 

29 

6 

958850 

29 

6 

961706 

28 

7 

953131 

34 

7 

956024 

34 

7 

958898 

34 

7 

961753 

33 

8 

953180 

39 

8 

956072 

38 

8 

958946 

38 

8 

961801 

38 

9 

L 

953228 

44 

9  i  956120 

43 

.   9 

958994 

43 

9 

961848 

43 

LOGARITHMS    OF   NUMBERS. 


37 


No. 

Log. 

Prop. 
Part. 

No. 

K*   iS£ 

No. 

Log. 

ss- 

No. 

L 

Log. 

Prop. 
Part 

1 

9160 

961895 

9220 

964731 

9280 

967548 

9340 

970347 

1 

961943 

5 

1 

964778 

5 

1 

967595 

5 

1 

970393 

5 

2 

961990 

10 

2 

964825 

9 

2 

967642 

9 

2 

970440 

9 

3 

962038 

14 

3 

964872 

14 

3 

967688 

14 

3 

970486 

14 

4 

962085 

19 

4 

964919 

19 

4 

967735 

19 

4 

970533 

19 

5 

962132 

24 

5 

964966 

24 

5 

967782 

23 

5 

970579 

23 

6 

962180 

28 

6 

965013 

28 

6 

967829 

28 

6 

970626 

28 

7 

962227 

33 

7 

965060 

33 

7 

967875 

33 

7 

970672 

33 

8 

962275 

38 

8 

965108 

38 

8 

967922 

38 

8 

970719 

37 

9 

962322 

43 

9 

965155 

42 

9 

967969 

42 

9 

970765 

42 

9170 

962369 

9230 

965202 

9290 

968016 

9350 

970812 

1 

962417 

5 

1 

965249 

5 

1 

968062 

5 

1 

970858 

5 

2 

962464 

9 

2 

965296 

9 

2 

968109 

9 

2 

970904 

9 

3 

962511 

14 

3 

965343 

14 

3 

968156 

14 

3 

970951 

14 

4 

962559 

19 

4 

965390 

19 

4 

968203 

19 

4 

970997 

19 

5 

962606 

24 

5 

965437 

24 

5 

968249 

23 

5 

971044 

23 

6 

962653 

28 

6 

965484 

28 

6 

968296 

28 

6 

971090 

28 

7 

962701 

33 

7 

965531 

33 

7 

968343 

33 

7 

971137 

33 

8 

962748 

38 

8 

965578 

38 

8 

968389 

38 

8 

971183 

37 

9 

962795 

42 

9 

965625 

42 

9 

968436 

42 

9 

971229 

42 

9180 

962843 

9240 

965672 

9300 

968483 

9360 

971276 

1 

962890 

5 

1 

965719 

5 

1 

968530 

5 

1 

971322 

5 

2 

962937 

9 

2 

965766 

9, 

2 

968576 

9 

2 

971369 

9 

3 

962985 

14 

3 

965813 

14 

3 

968623 

14 

3 

971415 

14 

4 

963032 

19 

4 

965860 

19 

4 

968670 

19 

4 

971461 

19 

5 

963079 

24 

5 

965907 

24 

5 

968716 

23 

5 

971508 

23 

6 

963126 

28 

6 

965954 

28 

6 

968763 

28 

6 

971554 

28 

7 

963174 

33 

7 

966001 

33 

7 

968810 

33 

7 

971600 

33 

8 

963221 

38 

8 

966048 

38 

8 

968856 

37 

81971647 

37 

9 

963268 

42 

9 

966095 

42 

9 

968903 

42 

9 

971693 

42 

9190 

963315 

9250  966142 

9310  968950 

9370 

971740 

1 

963363 

5 

1 

966189 

5 

1 

968996 

5 

1 

971786 

5 

2- 

963410 

9 

2 

966236 

9 

2 

969043 

9 

2 

971832 

9 

3 

963457 

14 

3 

966283 

14 

3 

969090 

14 

3 

971879 

14 

4 

963504 

19 

4 

966329 

19 

4 

969136 

19 

4 

971925 

19 

5 

963552 

24 

5 

966376 

24 

5 

969183 

23 

5 

971971 

23 

6 

963599 

28 

6 

966423 

28. 

6 

969229 

28 

6 

972018 

28 

7 

963646 

33 

7 

966470 

33 

7 

969276 

33 

7 

972064 

33 

8 

963693 

38 

8 

966517 

38 

8 

969323 

37 

8 

972110 

37 

9 

963741 

42 

9 

966564 

42 

9 

969369 

42 

9 

972156 

42 

9200 

963788 

9260 

966611 

9320 

969416 

9380 

972203 

1 

963835 

5 

1 

966658 

5 

1 

969462 

5 

1 

972249 

5 

2 

963882 

9 

2 

966705 

9 

2 

969509 

9 

2 

972295 

9 

3 

963929 

14 

3 

966752 

14 

3 

969556 

14 

3 

972342 

14 

4 

963977 

19 

4 

966798 

19 

4 

969602 

19 

4 

972388 

18 

5 

964024 

24 

5 

966845 

24 

5 

969649 

23 

5 

972434 

23 

6 

964071 

28 

6 

966892 

28 

6 

969695 

28 

6 

972480 

28 

7 

964118 

33 

7 

966939 

33 

7 

969742 

33 

7 

972527 

32 

8 

964165 

38 

8 

966986 

38 

8 

969788 

37 

8 

972573 

37 

9 

964212 

42 

9 

967033 

42 

9 

969835 

42 

9 

972619 

41 

9210 

964260 

9270 

967080 

9330 

969882 

9390 

972666 

1 

964307 

5 

1 

967127 

5 

1 

969928 

5 

1 

972712 

5 

2 

964354 

9 

2 

967173 

9 

2 

969975 

9 

2 

972758 

9 

3 

964401 

14 

3 

967220 

14 

3 

970021 

14 

3 

972804 

14 

4 

964448 

19 

4 

967267 

19 

4 

970068 

19 

4 

972851 

18 

5 

964495 

24 

5 

967314 

24 

5 

970114 

23 

5 

972897 

23 

6 

964542 

28  , 

6 

967361 

28 

6 

970161 

28 

6 

972943 

28 

7 

964590 

33 

7 

967408 

33 

7 

970207 

33 

7 

972989 

32 

8 

964637 

88 

8 

967454 

38 

8 

970254 

37 

8 

973035 

37 

9 

964684 

42 

9 

967501 

42 

9 

970300 

42 

9 

973082 

41 

38 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop.  | 
Part. 

9400 

973128 

9460 

975891 

9520 

978637 

9580 

981365 

1 

973174 

5 

1 

975937 

5 

1 

978683 

5 

1 

981411 

5 

2 

973220 

9 

2 

975983 

9 

2 

978728 

9 

2 

981456 

9 

3 

973266 

14 

3 

976029 

14 

3 

978774 

14 

3 

981501 

14 

4 

973313 

18 

4 

976075 

18 

4 

978819 

18 

4 

981547 

18 

5 

973359 

23 

5 

976121 

23 

5 

978865 

23 

5 

981592 

23 

6 

973405 

28 

6 

976166 

28 

6 

978911 

27 

6 

981637 

27 

7 

973451 

32 

7 

976212 

32 

7 

978956 

32 

7 

981683 

32 

8 

973497 

37 

8 

976258 

37 

8 

979002 

36 

8 

981728 

36 

9 

973543 

41 

9 

976304 

41 

9 

979047 

41 

9 

981773 

41 

9410 

973590 

9470 

976350 

9530 

979093 

9590 

981819 

1 

973636 

5 

1 

976396 

5 

1 

979138 

5 

1 

981864 

5 

2 

973682 

9 

2 

976442 

9 

2 

979184 

9 

2 

981909 

9 

3 

973728 

14 

3 

976487 

14 

3 

979230 

14 

3 

981954 

14 

4 

973774 

18 

4 

976533 

18 

4 

979275 

18 

4 

982000 

18 

5 

973820 

23 

5 

976579 

23 

5 

979321 

23 

5 

982045 

23 

6 

973866 

28 

6 

976625 

28 

6 

979366 

27 

6 

982090 

27 

7 

973913 

32 

7 

976671 

32 

7 

979412 

32 

7 

982135 

32 

8 

973959 

37 

8 

976717 

37 

8 

979457 

36 

8 

982181 

36 

9 

974005 

41 

9 

976762 

41 

9 

979503 

41 

9 

982226 

41 

9420 

974051 

9480 

976808 

9540 

979548 

9600 

982271 

1 

974097 

5 

1 

976854 

5 

1 

979594 

5 

1 

982316 

5  ' 

2 

974143 

9 

2 

976900 

9« 

2 

979639 

9 

2 

982362 

9 

3 

974189 

14 

3 

976946 

14 

3 

979685 

14 

3 

982407 

14 

4 

974235 

18 

4 

976991 

18 

4 

979730 

18 

4 

982452 

18 

5 

974281 

23 

5 

977037 

23 

5 

979776 

23 

5 

982497 

23 

6 

974327 

28 

6 

977083 

27 

6 

979821 

27 

6 

982543 

27 

7 

974373 

32 

7 

977129 

32 

7 

979867 

32 

7 

982588 

32 

8 

974420 

37 

8 

977175 

37 

8 

979912 

36 

8 

982633 

36 

9 

974466 

41 

9 

977220 

41 

9 

979958 

41 

9 

982678 

41 

9430 

974512 

9490 

977266 

9550 

980003 

9610 

982723 

1 

974558 

5 

1 

977312 

5 

1 

980049 

5 

1 

982769 

5 

2 

974604 

9 

2 

977358 

9 

2 

980094 

9 

2 

982814 

9 

3 

974650 

14 

3 

977403 

14 

3 

980140 

14 

3 

982859 

14 

4 

974696 

18 

4 

977449 

18 

4 

980185 

18 

4 

982904 

1^ 

5 

974742 

23 

5 

977495 

23 

5 

980231 

23 

5 

982949 

23 

6 

974788 

28 

6 

977541 

27' 

6 

980276 

27 

6 

982994 

27 

7 

974834 

32 

7 

977586 

32 

7 

980322 

32 

7 

983040 

32 

8 

974880- 

37 

8 

977632 

37 

8 

980367 

36 

8 

983085  36 

9 

974926 

41 

9 

977678 

41 

9 

980412 

41 

9 

983130  41 

9440 

974972 

9500 

977724 

9560 

980458 

9620 

983175 

1 

975018 

5 

1 

977769 

5 

1 

980503 

5 

1 

983220  1  5 

2 

975064 

9 

2 

977815 

9 

2 

980549 

9 

2 

983265 

9 

3 

975110 

14 

3 

977861 

14 

3 

980594 

14 

3 

983310 

14 

4 

975156 

18 

4 

977906 

18 

4 

980640 

18 

4 

983356 

18 

5 

975202 

23 

5 

977952 

23 

5 

980685 

23 

5 

983401 

23 

6 

975248 

28 

6 

977998 

27 

6 

980730 

27 

6 

983446 

27 

7 

975294 

32 

7 

978043 

32 

7 

980776 

32 

7 

983491 

32 

8 

975340 

37 

8 

978089 

37 

8 

980821 

36 

8 

983536 

36 

9 

975386 

41 

9 

978135 

41 

9 

980867 

41 

9 

983581 

41 

9450 

975432 

9510 

978180 

9570 

980912 

9630 

983626 

1 

975478 

5 

1 

978226 

5 

1 

980957 

5 

1 

983671 

5 

-   2 

975524 

9 

2 

978272 

9 

2 

981003 

9 

2 

983716 

9 

3 

975570 

14 

3 

978317 

14 

3 

981048 

14 

3 

983762 

14 

4 

975616 

18 

4 

978363 

18 

4 

981093 

18 

4 

983807 

18 

5 

975661 

23 

5 

978409 

23 

5 

981139 

23 

5 

983852 

23 

6 

975707 

28 

6 

978454 

27 

6 

981184 

27 

6 

983897 

27 

7 

975753 

32 

7 

978500 

32 

7 

981229 

32 

7 

983942 

32 

8 

975799 

37 

8 

978546 

37 

8 

981275 

36 

8  983987 

36 

9 

975845 

41 

9 

978591 

41 

9 

981320 

41 

9  1  984032 

41 

LOGARITHMS    OF   NUMBERS. 


No. 

*">*  S: 

No. 

Log. 

Prop. 
Part. 

No. 

Log 

^"1'-:   No/ 

Part. 

^-  pa,1:; 

9640 

984077 

9700 

986772 

9760 

989450 

9820 

992111 

1 

984122 

5 

1 

986816 

4 

1 

989494 

4 

1 

992156  |  4 

2 

984167 

9 

2 

986861 

9 

2 

989539 

9 

2 

992200   9 

3 

984212  14 

3 

986906 

13 

3 

989583 

13     3 

992244  13 

4 

984257 

18 

4 

986951 

18 

4  989628 

18     4 

992288  18 

5 

984302 

23 

5 

986995 

22 

5 

989672 

22  i    5 

992333  22 

6 

984347 

27 

6 

987040  27 

6 

989717 

27     6 

992377  26 

7 

984392 

32 

7 

987085  31 

7 

989761 

31 

7 

992421 

31 

8 

984437  36 

8 

987130  36     8  989806 

36 

8 

992405 

35 

9 

984482 

41 

9 

987174 

40 

9  989850 

40 

9 

9i#dt)9 

40 

9650 

984527 

9710 

987219 

9770 

989895 

9830 

992553 

1 

984572 

5 

1 

987264 

4 

1 

989939 

4 

1 

992598 

4 

2 

984617 

9 

2 

987309 

9 

2 

989983 

9  !    2 

992042 

9 

3 

984662 

14 

3 

987353 

13  j 

3 

990028 

13     3 

992686 

13 

4 

984707 

18 

4 

987398 

18 

4 

990072 

18     4 

992730 

18 

5 

984752 

23 

5 

987443 

22 

5 

990117 

22 

5 

992774 

22 

6 

984797 

27 

6 

987487 

27 

6 

990161 

27 

6 

992818 

26 

7 

984842 

32 

7 

987532 

31 

7 

990206 

31 

7 

9U2863 

31 

8 

984887 

36 

.   8 

987577 

36 

8 

990250 

36 

8 

992907 

35 

9 

984932 

41 

9 

987622 

40 

9 

990294 

40 

9 

992951 

40 

9660 

984977 

9720 

987666 

9780 

990339 

9840 

992995 

1 

985022 

5 

1 

987711 

4 

1 

990383 

4 

1  j  993039 

4 

2 

985067 

9 

2 

987756 

9 

2 

990428 

9 

2 

993083 

9 

3 

985112 

14 

3-  987800 

13  i 

3 

990472 

13 

3 

993127 

13 

4 

985157 

18 

4  987845 

18 

4 

990516 

18 

4 

993172 

18 

5 

985202 

23 

5  987890 

22 

5 

990561 

22 

5 

993216 

22 

6 

985247 

27 

6 

98793-1 

27 

6 

990605 

27 

6 

993260 

26 

7 

985292 

32 

7  987979  31  !   7 

990650 

31 

7 

993304  31 

8 

985337 

36 

8  988024  36  I   8  990694 

36 

8 

993348 

35 

9 

985382 

41 

9  988068  40  I!   9  990738 

40 

9 

993392 

40 

9670 

985426 

97301988113     1  9790  990783 

9850 

993436 

1 

985471 

4 

1  1  988157   41    1  !  990827 

4 

1 

993480 

4 

2 

985516 

9 

2  i  988202   9  |    2  j  990871 

9 

2 

993524 

9 

3 

985561 

13 

3  988247 

13  f   3  990916 

13 

3 

993568 

13 

4  j  985606 

18 

4  988291 

18  |    4  990960 

18 

4 

993613 

18 

5 

985651 

22 

5 

988336 

22     5 

991004 

22 

5 

993657 

2'' 

6 

985696 

27 

6 

988381 

27  i    6 

991049 

27 

6 

993701 

26 

7 

985741 

31 

7 

988425 

31     7 

991093 

31 

7 

993745 

31 

8 

98578(3 

36 

8 

988470 

36 

8 

991137 

36   -  8 

993789 

35 

9 

985830 

40 

9 

988514 

40 

9 

991182 

40 

9 

993833 

40 

9680 

985875 

9740 

988559 

9800 

991226 

9860 

993877 

1 

985920 

4 

1 

988603   4 

1 

991270 

4 

1 

993921 

4 

2 

985965 

9 

2 

988648 

9 

2 

991315 

9 

2 

993965 

9 

3 

986010 

13 

3 

988693 

13 

3 

991359 

13 

3 

994009 

13 

4 

986055 

18 

4 

988737 

18 

4 

991403 

18 

4 

994053 

18 

5 

986100 

22 

5 

988782 

22 

5 

991448 

22 

5 

994097 

22 

6 

986144 

27 

6 

988826 

27 

6 

991492 

27 

6 

994141 

26 

7 

986189 

31 

7 

988871 

31 

>T 

991536 

31 

7 

994185 

31 

8 

986234 

36 

8 

988915 

36  i    8 

991580 

36 

8 

994229 

35 

9 

986279 

40 

9 

988960 

40     9 

991625 

40 

g 

994273 

40 

9690 

986324 

9750 

989005 

9810 

991669 

9870 

994317 

1 

986369 

4 

1 

989049 

4 

1  1991713 

4 

1 

994361 

4 

2  !  986413 

9 

2 

989094 

9  i    2 

991757 

9 

2  1  994405 

9 

3  986458 

13 

3 

989138 

13 

3 

991802 

13 

3 

994449 

13 

4  j  986503 

18 

4 

989183 

18 

4 

991846 

18 

4 

994493 

18 

5  986548 

22 

5 

989227 

22 

5 

991890 

22 

5 

994537 

22 

6 

986593 

27 

6 

989272 

27 

6 

991934 

27 

6 

994581 

26 

7 

986637 

31 

7 

989316 

31 

7 

991979 

31 

7 

994625 

31 

8 

986682 

36 

8 

989361 

36 

8 

992023 

36 

8 

994669  35 

9 

986727 

40 

9 

989405 

40 

9  992067 

40 

9 

994713'  40 

40 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

NO. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 

Part. 

9880 

994757 

9910 

996074 

9940 

997386 

9970 

998695 

1 

994801 

4 

1 

996117 

4 

1 

997430 

4 

1 

998739 

4 

2 

994845 

9 

2 

996161 

9 

2 

997474 

9 

2 

998782 

9 

3 

994889 

13 

3 

996205 

13 

3 

997517 

13 

3 

998826 

13 

4 

994933 

18 

4 

996249 

18 

4 

997561 

17 

4 

998869 

17 

5 

994977 

22 

5 

996293 

22 

5 

997605 

22 

5 

998913 

22 

6 

995021 

26 

6 

996336 

26 

6 

997648 

26 

6 

998956 

26 

7 

995064 

31 

7 

996380 

31 

7 

997692 

30 

7 

999000 

30 

8 

995108 

35 

8 

996424 

35 

8 

997736 

35 

8 

999043 

35 

9 

995152 

40 

9 

996468 

40 

9 

997779 

39 

9 

999087 

39 

9890 

995196 

9920 

996512 

9950 

997823 

9980 

999130 

1 

995240 

4 

1 

996555 

4 

1 

997867 

4 

1 

999174 

4 

2 

995284 

9 

2 

996599 

9 

2 

997910 

9 

2 

999218 

9 

3 

995328 

13 

3 

996643 

13 

3 

997954 

13 

3 

999261 

13 

4 

995372 

18 

4 

996687 

18 

4 

997998 

17 

4 

999305 

17 

5 

995416 

22 

5 

996730 

22 

5 

998041 

22 

5 

999348 

22 

6 

995460 

26 

6 

996774 

26 

6 

998085 

26 

6 

999392 

26 

7 

995504 

31 

7 

996818 

31 

7 

998128 

30 

7 

999435 

30 

8 

995547 

35 

8 

996862 

35 

8 

998172 

35. 

8 

999478 

35 

9 

995591 

40 

9 

996905 

40 

9 

998216 

39 

9 

999522 

39 

9900 

995635 

9930 

996949 

9960 

998259 

9990 

999565 

1 

995679 

4 

1 

996993 

4 

1 

998303 

4 

1 

999609 

4 

2 

995723 

9 

o 

997037 

9 

2 

998346 

9 

2 

999652 

9 

3 

995767 

13 

3 

997080 

13 

3 

998390 

13 

3 

999696 

13 

4 

995811 

18 

4 

997124 

18 

4 

998434 

17 

4 

999739 

17 

5 

995854 

22 

5 

997168 

22 

5 

998477 

22 

5 

999783 

22 

6 

996898 

26 

6 

997212 

26 

6 

998521 

26 

6 

999826 

26 

7 

995942 

31 

7 

997255 

31 

7 

998564 

30 

7 

999870 

30 

8 

995986 

35 

8 

997299 

35 

8  |  998608 

35 

8 

999913 

35 

9 

996030 

40 

9 

997343 

39 

9  i  998652 

39 

9 

999957 

39 

LOGAKITHMS  TO  50  DECIMAL  PLACES. 


0-00000000000000000000000000000000000000000000000000 
0-30102999566398119521373889472449302676818988146211 
0-47712125471966243729502790325511530920012886419069 
0-60205999132796239042747778944898605353637976292422 
0-69897000433601880478626110527550697323181011853789 

0-77815125038364363250876679797960833596831874565280 
0-84509804001425683071221625859263619348357239632397 
0-90308998699194358564121668417347908030456964438633 
0-95424250943932487459005580651023061840025772838139 
1-00000000000000000000000000000000000000000000000000 
1-04139268515822504075019997124302424170670219046645 
1-07918124604762482772250569270410136273650862711491 
1-11394335230683676920650515794232843082972918838707 
1-14612803567823802592595515331712922025176227778607 
1-17609125905568124208128900853062228243193898272859 

1-20411998265592478085495557889797210707275952584843 
1-23044892137827392854016U89432833703000756737842505 
1-25527250510330606980379470123472364516844760984350 
1-27875360095282896153633347575692931795112933739450 
1-30102999566398119521373889472449302676818988146211 

1-32221929473391926800724416184775150268370126051866 
1-34242268082220623596393886596751726847489207192856 
1-36172783601759287886777711225118954969751103433610 
1-38021124171160602293624458742859438950469850857702 
1-39794000867203760957252221055101394646362023707578 


LOGARITHMIC    SINES,    ETC. 


41 


0  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Cosine. 

' 

0] 

Infinite. 

Infinite. 

000000 

10-OOOOOOj 

60  I 

1 

6-463726 

3-536274 

6-463726 

13-536274 

000000 

10-000000 

59 

21 

6-764756 

501717 

3-235244  6-764756 

501717 

13-235244 

000000 

10-000000 

58 

3 

6-940847 

293485 

3-059153  6-940847 

293485 

13-059153 

000000 

10-000000 

57 

4 

7-065786 

208231 

2-934214|7-065786 

208231 

12-934214 

000000 

10-000000 

56 

5 

7-162696 

161517 

2-837304!:7-162696 

161517 

12-837304 

000000 

10-000000 

55 

6 

7-241877 

131968 

2-758123'  7-241878 

131969 

12-758122 

000001 

9-999999 

54 

7 

7-308824 

111578 

2-691176  7-308825 

111578 

12-691175 

000001 

9-999999 

53 

8 

7-366816 

96653 

2-633184  7-366817 

96653 

12-633183 

000001 

9-999999 

52 

9 

7-417968 

85254 

2-582032  7-417970 

85254 

12-582030 

000001 

9-999999 

51 

10 

7-463726 

76262 

2-536274 

7-463727 

76263 

12-536273 

000002 

9-999998  50  | 

11 

7-505118 

68988 

2-494882  7-505120 

68988 

12-494880 

000002 

9-999998 

49 

12 

7-542906 

62981 

2-457094IJ7-542909 

62981 

12-457091 

000003 

9-999997 

48 

13  7-577668 

57936 

2  -422332  |7  -577672 

57937 

12-422328 

000003 

9-999997 

47 

14 

7-609853 

53641 

2-390147  7-609857 

53642 

12-390143 

000004 

9-999996 

46 

15 

7-639816 

49938 

2-360184 

7-639820 

49939 

12-360180 

000004 

9-999996 

45 

16 

7-667845 

46714 

2-332155! 

7-667849 

46715 

12-332151 

000005 

9-999995 

44 

17 

7-694173 

43881 

2-305827 

7-694179 

43882 

12-305821 

000005 

9-999995 

43 

18 

7-718997 

41372 

2-2810031 

7-719003 

41373 

12-280997 

000006 

9-999994 

42  1 

19 

7-742478 

39135 

2-257522i 

7-742484 

39136 

12-257516 

•000007 

9-999993 

41 

20 

,7-764754 

37127 

2-235246 

7-764761 

37128 

12-235239 

•000007 

9-999993 

40 

21  7-785943 

35315 

2-214057 

7-785951 

35315 

12-214049 

•000008 

9-999992 

39 

22 

7-806146 

33672 

2-193854! 

7-806155 

33673 

12-193845 

•000009 

9-999991 

38 

23  |7  -825451 

32175 

2-174549 

7-825460 

32176 

12-174540 

•000010 

9-999990 

37 

24 

7-843934 

30805 

2-156066  7-843944 

30807 

12-156056 

•000011 

9-999989 

36 

25i  7-861662 

29547 

2-138338  ]7  -861674 

29549 

12-138326 

•000011 

9-999989 

35 

26 

7-878695 

28388 

2-121305;  7-878708 

28390 

12-121292 

•000012 

9-999988 

34 

27!|7-895085 

27317 

2-104915  7-895099 

27318 

12-104901 

•000013 

9-999987 

33 

28  17-910879 

26323  2-089121  7-910894 

26325 

12-089106 

•000014 

9-999986 

32 

29 

7-926119 

25399  2-073881  117-926134 

25401 

12-073866 

•000015 

9-999985 

31 

30 

7-940842 

24538  2-059158  17-940858 

24540 

12-059142 

•000017  9-999983 

30 

31 

7-955082 

23733  2-044918  17-955100 

23735 

12-044900 

•000018  9-999982 

29 

32 

7-968870 

22980 

2-031130 

7  -968889 

22982 

12-031111 

•000019  9-999981 

28 

33 

7-982233'  22273 

2-017767 

7-982253 

22275 

12-017747 

•0000201  9-999980 

27 

34 

7-995198  21608 

2-004802 

7-995219 

21610|12-004781 

•000021  1  9-999979 

26 

35 

8-007787  20981 

1-992213 

8-007809 

2098311-992191 

•0000231  9-999977 

25 

36 

'8-020021  20390 

i  -979979 

8-020045 

20392 

11-979955 

•000024  9-999976 

24 

37 

8-031919  .19831 

1-968081 

8-031945 

19833 

11-968055 

•0000251  9-999975 

23 

38 

8-043501  19302 

1-956499 

8-043527 

19305 

11-956473 

•000027  9-999973 

22 

39 

8-054781  18801 

1-945219 

8-054809 

18803 

11-945191 

•000028  9-999972 

21 

40 

'8-065776  18325 

1-934224 

8-065806 

18327 

11-934194 

•000029  9-999971 

20 

41 

8-076500  17872 

1-923500 

8-076531 

17875 

11-923469 

•000031 

9-999969 

19 

42 

8-086965  17441 

1-913035 

8-086997 

17444 

11-913003 

•000032 

9-999968 

18 

43 

8-097183  17031 

1-902817 

8-097217 

17034 

11-902783 

•000034 

9-999966 

17 

44 

18-107167  16639 

1-892833 

8-107202 

16642 

11-892798 

•000036 

9-999964 

16 

45 

8-116926  16265 

1-883074 

8-116963 

16268 

11-883037 

•000037 

9-999963 

15 

46 

8-r26471  15908 

1-873529 

8-126510 

15911 

11-873490 

•000039 

9-999961 

14 

47 

8-135810  15566 

1-864190 

8-135851 

15568 

11-864149 

•000041 

9-999959 

13 

48 

8-144953  15238 

1  -855047 

8-144996 

15241 

11-855004 

•000042 

9-999958 

12 

49 

8-153907  14924 

1-846093 

8-153952 

14927 

11-846048 

•000044 

9-999956 

11 

50 

8-162681  14622 

1-837319 

8-162727 

14625 

11-837273 

•000046 

9-999954 

10 

51 

8-171280  14333 

1-828720 

8-171328 

14336 

11-328672 

•000048 

9-999952 

9 

52 

8-179713  14054 

1-820287 

8-179763 

14057 

11-820237 

•000050 

9-999950 

8 

53  8-187985  13786 

1-812015  8-188036 

13790 

11-811964 

•000052 

9-999948 

7  i 

548-196102  13529 

1-803898  8-196156 

13532 

11-803844 

•000054 

9-999946 

6 

55|!8-204070  13280 

1-795930  8-204126 

13284 

11-795874 

•000056  9-999944 

5 

66  8-211895  13041 

1-788105;  8-211953 

13044 

11-788047 

•000058|  9-999942 

4 

57 

8-219581:  12810 

1-780419  "8-219641 

12814 

11-780359 

•000060!  9-999940 

3  I 

588-227134  12587 

1-712866  8-227195 

12591 

11-772805-0000621  9-999938 

2 

598-234557  12372 

1-765448  8-234621 

12376 

11-765379 

•000064  9-999936 

1  i 

60!  8-241855  12164 

1-758145  8-241922J  12168 

11-758078 

•000066  9-999934 

o  i 

' 

Cosine. 

Secant.  :  C  'tangent. 

Twigeat 

Cosecant.  '    Sine. 

/  1 

40 


LOGARITHMS    OF   NUMBERS. 


No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

No. 

Log. 

Prop. 
Part. 

9880 

994757 

9910 

996074 

9940 

997386 

9970 

998695 

1 

994801 

4 

1 

996117 

4 

1 

997430 

4 

1 

998739 

4 

2 

994845 

9 

2 

996161 

9 

2 

997474 

9 

2 

998782 

9 

3- 

994889 

13 

3 

996205 

13 

3 

997517 

13 

3 

998826 

13 

4 

994933 

18 

4 

996249 

18 

4 

997561 

17 

4 

998869 

17 

5 

994977 

22 

5 

996293 

22 

5 

997605 

22 

5 

998913 

22 

6 

995021 

26 

6 

996336 

26 

6 

997648 

26 

6 

998956 

26 

7 

995064 

31 

7 

996380 

31 

7 

997692 

30 

7 

999000 

30 

8 

995108 

35 

8 

996424 

35 

8 

997736 

35 

8 

999043 

35 

9 

995152 

40 

9 

996468 

40 

9 

997779 

39 

9 

999087 

39 

9890 

995196 

9920 

996512 

9950 

997823 

9980 

999130 

1 

995240 

4 

1 

996555 

4 

1 

997867 

4 

1 

999174 

4 

2 

995284 

9 

2 

996599 

9 

2 

997910 

9 

2 

999218 

9 

3 

995328 

13 

3 

996643 

13 

3 

997954 

13 

3 

999261 

13 

4 

995372 

18 

4 

996687 

18 

4 

997998 

17 

4 

999305 

17 

5 

995416 

22 

5 

996730 

22 

5 

998041 

22 

5 

999348 

22 

6 

995460 

26 

6 

996774 

26 

6 

998085 

26 

6 

999392 

26 

7 

995504 

31 

7 

996818 

31 

7 

998128 

30 

7 

999435 

30 

8 

995547 

35 

8 

996862 

35 

8 

998172 

35. 

8 

999478 

35 

9 

995591 

40 

9 

996905 

40 

9 

998216 

39 

9 

999522 

39 

9900 

995635 

9930 

996949 

9960 

998259 

9990 

999565 

1 

995679 

4 

1 

996993 

4 

1 

998303 

4 

1 

999609 

4 

2 

995723 

9 

O 

997037 

9 

2 

998346 

9 

2 

999652 

9 

3 

995767 

13 

8 

997080 

13 

3 

998390 

13 

3 

999696 

13 

4 

995811 

18 

4 

997124 

18 

4 

998434 

17 

4 

999739 

17 

5 

995854 

22 

5 

997168 

22 

5 

998477 

22 

5 

999783 

22 

6 

996898 

26 

6 

997212 

26 

6 

998521 

26 

6 

999826 

26 

7 

995942 

31 

7 

997255 

31 

7 

998564 

30 

7 

999870 

30 

8 

995986 

35 

8 

997299 

35 

8 

998608 

35 

8 

999913 

35 

9 

996030 

40 

9 

997343 

39 

9 

998652 

39 

9 

999957 

39 

LOGARITHMS  TO  50  DECIMAL  PLACES. 


0-00000000000000000000000000000000000000000000000000 
0-30102999566398119521373889472449302676818988146211 
0-47712125471966243729502790325511530920012886419069 
0-60205999132796239042747778944898605353637976292422 
0-69897000433601880478626110527550697323181011853789 
0-77815125038364363250876679797960833596831874565280 
0-845098040014256830712216258592636193483572396-32397 
0-90308998699194358564121668417347908030456964438633 
0-95424250943932487459005580651023061840025772838139 
1-00000000000000000000000000000000000000000000000000 
1-04139268515822504075019997124302424170670219046645 
1-07918124604762482772250569270410136273650862711491 
1-11394335230683676920650515794232843082972918838707 
1-14612803567823802592595515331712922025176227778607 
1-17609125905568124208128900853062228243193898272859 

1-20411998265592478085495557889797210707275952584843 
1-23044892137827392854016989432833703000756737842505 
1-25527250510330606980379470123472364516844760984350 
1-27875360095282896153633347575692931795112933739450 
1-30102999566398119521373889472449302676818988146211 

1-32221929473391926800724416184775150268370126051866 
1-34242268082220623596393886596751726847489207192856 
1-36172783601759287886777711225118954969751103433610 
1-38021124171160602293624458742859438950469850857702 
1-39794000867203760957252221055101394646362023707578 


LOGARITHMIC    SINES,    ETC 


41 


0  DEO. 


'1 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Cosine. 

i 

<J 

Infinite. 

Infinite. 

000000 

10-OOOOOOi 

60 

1 

6-463726 

3-536274 

6-463726 

13-536274 

000000 

10-000000 

59 

21 

6-764756 

501717 

3-235244  6-764756 

501717 

13-235244 

000000 

10-000000 

58 

3 

6-940847 

293485 

3-059153  ,6-940847 

293485 

13-059153 

000000 

10-000000 

57 

4 

7-065786 

208231 

2-934214  7-065786 

208231 

12-934214 

000000 

10-000000 

56 

5 

7-162696 

161517 

2-837304  i7-162696 

161517 

12-837304 

000000 

10-OOOOOOi 

55 

6 

7-241877 

131968 

2-758123'  7-241878 

131969 

12-758122 

000001 

9-999999 

54 

7 
8 

7-308824 
7-366816 

111578 
96653 

2-691176  17-308825 
2-633184  7-366817 

111578 
96653 

12-691175 
12-633183 

•000001 
•000001 

9-999999 
9-999999 

53 
52 

9 

7-417968 

85254 

2-582032  7-417970 

85254 

12-582030 

•000001 

9-999999 

51 

10 

7-463726 

76262 

2-536274  7-463727 

76263 

12-536273 

•000002 

9-999998 

50 

11 

7-505118 

68988 

2-494882  7-505120 

68988 

12-494880 

•000002 

9-999998 

49 

121 

7-542906 

62981 

2-457094  17-542909 

62981 

12-457091 

•000003 

9-999997 

48  i 

13! 

7-577668 

57936 

2-422332 

7-577672 

57937 

12-422328 

•000003 

9-999997 

47 

14l  7-609853 

53641 

2-390147 

7-609857 

53642 

12-390143 

•000004 

9-999996 

46 

15  7-639816 

49938 

2-360184 

7-639820 

49939 

12-360180 

•000004 

9-999996 

45 

10 

7-667845 

46714 

2-332155 

7-667849 

46715 

12-332151 

•000005  9-999995 

44 

17 

7-694173 

43881 

2-305827 

7-694179 

43882 

12-305821 

•000005  9-999995 

43 

18! 

7-718997 

41372 

2-281003!7-719003 

41373 

12-280997 

•000006 

9-999994 

42 

19j 

7-742478 

39135 

2-25752217  -742484 

39136 

12-257516 

•000007 

9-999993 

41 

20 

7-764754 

37127 

2-235246 

7-764761 

37128 

12-235239 

•000007 

9-999993 

40 

2117-785943 

35315 

2-214057! 

7-785951 

35315 

12-214049 

•000008 

9-999992 

39 

22!  7-806146 

33672 

2-193854: 

7-806155 

33673 

12-193845 

•000009 

9-999991 

38 

23j  7  -825451 

32175 

2-174549 

7-825460 

32176 

12-174540 

•000010 

9-999990 

37 

24J7-843934 

30805 

2-156066 

7-843944 

30807 

12-156056 

•000011 

9-999989 

36 

25 

7-861662 

29547 

2-138338 

7-861674 

29549 

12-138326 

•000011 

9-999989 

35 

26 

7-878695 

28388 

2-121305 

7-878708 

28390 

12-121292 

•000012 

9-999988 

34 

27 

7-895085 

27317 

2-104915  7-895099 

27318 

12-104901 

•000013 

9-999987 

33 

28 

7-910879 

26323 

2-089121  7-910894 

26325 

12-089106 

•000014 

9-999986 

32 

2917-926119 

25399 

2-073881  7-926134 

25401 

12-073866 

•0000151  9-999985 

31 

30-7-940842 

24538 

2-059158  17-940858 

24540 

12-059142 

•000017  9-999983 

30 

31 

7-955082 

23733 

2-044918:7-955100 

23735 

12-044900 

•000018 

9-999982 

29 

32 

7-968870 

22980 

2-031  1301  7  -968889 

22982 

12-031111 

•000019  9-999981 

28 

33 

7-9822331  22273 

2-017767 

7-982253 

22275 

12-017747 

•000020  9-999980 

27 

34 

7-995198  21608 

2-004802 

7-995219 

21610  12-004781 

•000021  9-999979 

26 

35 

8-007787  20981 

1-992213 

8-007809 

20983  11-992191 

•000023  9-999977 

25 

36 

8-020021  20390 

1-979979 

8-020045 

20392 

11-979955 

•000024  9-999976 

24 

37 

8-031919  .19831 

1-968081 

8-031945 

19833 

11-968055 

•000025  9-999975 

23 

38 

8-043501  19302 

1-956499 

8-043527 

19305 

11-956473 

•000027  9-999973 

22 

39 

8-054781  18801 

1-945219 

8-054809 

18803 

11-945191 

•000028 

9-999972 

21 

40 

8-065776  18325 

1-934224 

8-065806 

18327 

11-934194 

•000029 

9-999971 

20 

41 

8-076500  17872 

1-923500 

8-076531 

17875 

11-923469 

•000031 

9-999969 

19 

42 

8-086965  17441 

1-913035 

8-086997 

17444 

11-913003 

•000032 

9-999968 

18 

43 

8-097183  17031 

1-902817 

8-097217 

17034 

11-902783 

•000034 

9-999966 

17 

44 

:8-107167  16639 

1-892833 

8-107202 

16642 

11-892798 

•000036 

9-999964 

16 

45 

8-116926  16265 

1-883074 

8-116963 

16268 

11-883037 

•000037 

9-999963 

15 

46 

8-r26471  15908 

1-873529 

8-126510 

15911 

11-873490 

•000039 

9-999961 

14 

47 

8-135810  15566 

1-864190 

8-135851 

15568 

11-864149 

•000041 

9-999959 

13 

48 

8-144953  15238 

1-855047 

8-144996 

15241 

11-855004 

•000042 

9-999958 

12 

49 

8-153907  14924 

1-846093 

8-153952 

14927 

11-846048 

•000044 

9-999956 

11 

50 

8-162681  14622 

1-837319 

8-162727 

14625 

11-837273 

•000046 

9-999954 

10 

51 

8-171280  14333 

1-828720 

8-171328 

14336 

11-828672 

•000048 

9-999952 

9 

52 
53 

8-179713  14054 
8-187985  13786 

1-820287  8-179763 
1-8120151  8-188036 

14057 
13790 

11-820237 
11-811964 

•000050 
•000052 

9-999950 
9-999948 

8 

7 

54  8-196102  13529 

1-803898 

8-196156 

13532 

11-803844 

•000054 

9-999946 

6 

5518-204070  13280 

1  -795930  i8-204126 

13284 

11-795874 

•000056 

9-999944 

5 

66|8-211895  13041 

1  -788105!  8-211953 

13044 

11-788047 

•000058  9-999942 

4 

57 

'8-219581!  12810 

1-780419  8-219641 

12814 

11-780359 

•000060  9-999940 

3 

588-227134  12587 

1-7^2866  8-227195 

12591 

11  -772805,  -000062  9-999938 

2 

5918-234557  12372 

1-765443  8-234621  12376 

11-7653791  -000064  9-999936 

1 

60 

8-241855  12164 

1-7581458-241922  12168 

11-758078 

•000066  9-999934 

0 

' 

|   Cosine. 

Secant.  !  Cotangent. 

Tangent. 

Cosecant.     Sine. 

' 

42              .  LOGARITHMIC  SINES,  ETC. 

1  DEC. 

'I  Si°" 

Diff. 

100" 

Cosecant. 

Tangent,  j   g«J- 

Cotangent. 

Cosine. 

' 

(!i;8-241855 

1-758145 

8-241921! 

11-758079 

•00006(5 

9-999934 

60 

Ir8-249033 

11963 

1-750967 

8-249102 

11967 

11-750898 

•000068 

9-999932 

59 

21  8-256094 

11768 

1  -743906 

8-256165 

11772 

11-743835 

•000071 

9-999929 

58 

3  !8-263042 

11580 

1-736958 

8-263115 

11584 

11-736885 

•000073 

9-999927 

57 

4 

8-269881 

11397 

1-730119 

8-269956 

11402 

11-730044 

•000075 

9-999925 

56 

5 

8-276614 

11221 

1  -723386 

8-276691 

11225 

11-723309 

•000078i  9-999922 

55 

6 

8-283243 

11050 

1-716757: 

8-283323 

11054 

11-710677 

•000080 

9-999920 

54 

7 

8-289773 

10883 

1-710227 

8-289856 

10887 

11-710144 

•000082 

9-999918 

53 

* 

8-296207 

10722 

1-703793 

8-296292 

10726 

11-703708 

•000085 

9-999915 

52 

9 

8-302546 

10565 

1-697454  18-302634 

10570 

11-697366 

•000087 

9-999913 

51 

10 

8-308794 

10413 

1-691206 

8-308884 

10418 

11-691116 

•000090 

9-999910 

50 

11 

8-314954 

10266 

1-685046! 

8-315046 

10270 

11-684954 

•000093 

9-999907 

49 

12 

iS-321027 

10122 

1-678973; 

8-321122 

10126 

11-678878 

-0000951  9-999905 

48 

13 

8-327016 

9982 

1-672984  8-327114 

9987 

11-672886 

•0000981  9-999902 

47 

14 

18-332924 

9847 

1-66707H 

8-333025 

9851 

11-666975 

•000101 

9-999899 

46 

15 

8-338753 

9714 

1-661247 

8-338856 

9719 

11-661144 

•000103 

9-999897 

45 

16 

8-344504 

9586 

1  -655496 

8-344610 

9590 

11-655390 

•000106 

9-999894 

44 

17 

8-350181 

9460 

1-649819 

8-350289 

9465 

11-649711 

•000109 

9-999891 

43 

18 

8-355783 

9338' 

1-644217 

8-355895 

9343 

11-644105 

•000112 

9-999888 

42 

19 

;8-361315 

9219 

1-638685 

8-361430 

9224 

11-638570 

•000115 

9-999885 

41 

20 

:8-366777 

9103 

1-633223 

8-366895 

9108 

11-633105 

•000118 

9-999882 

40 

21 

8-372171 

8990 

1-627829 

8-372292 

8995 

11-627708 

•000121 

9-999879 

39 

22 

8-377499 

8880 

1-622501 

8-377622 

8885 

11-622378 

•000124 

9-999876 

38 

23  8-382762 

8772 

1-617238 

8-382889 

8777 

11-617111 

•000127 

9-999873 

37 

24 

8-387962 

8667 

1-612038 

8-388092 

8672 

11-611908 

•000130 

9-999870 

36 

25 

8-393101 

8564 

1-606899 

8-393234 

8570 

11-606766 

•000133 

9-999867 

35 

26 

8-398179 

8464 

1-601821 

8-398315 

8470 

11-601685 

•000136)  9-999864 

34 

27 

8-403199 

8366 

1-596801 

8-403338 

8371 

11-596662 

•000139 

9-999861 

33 

28 

8-408161 

8271 

1-591839 

8-408304 

8276 

11-591696 

•000142 

9-999858 

32 

29 

8-413068 

8177 

1-586932 

8-413213 

8182 

11-586787 

•000146!  9-999854 

31 

30 

8-417919 

8086 

1-582081 

8-418068 

8091 

11-581932 

•000149 

9-999851 

30 

31H8-422717 

7996  1-577283 

8-422869 

8002 

11-577131 

•000152 

9-999848 

29 

32  8-427462 
33  8-432156 

7909|l-572538 
7823  1-567844 

8-427618  7914  11-572382 
8-432315  7829  11-567685 

•0001561  9-999844 
•000159)  9-999841 

28 
27 

34'  '8-436800  7740 

1-563200 

8-436962  7745  11-563038 

•000162 

9-999838 

26 

35J8-441394  7657 

1-558606: 

8^441560  7663111-558440 

•000166 

9-999834 

25 

36J8-445941  7577  1-554059 

8-446110 

7583|11-553890 

•000169  9-999831 

24 

37  8-450440  7499  1-549560 

8-450613 

7505  ill  -549387 

•000173 

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23 

3818-454893  7422  1  1-545107 

8-455070 

7428  111  -544930 

•000176 

9-999824 

22 

391  '8-459301!  7346 

1-540699 

8-459481 

7352 

11-540519 

•000180 

9-999820 

21 

40!  8-463665;  7273 

1-536335 

8-463849 

7279 

11-536151 

•000184J  9-999816 

20 

41H8-4679851  7200 

1-532015 

8-468172 

7206 

11-531828 

•000187  9-999813 

19 

42J8-472263J  7129 

1-527737 

8-472454 

7135 

11-527546 

•0001911  9-999809 

18 

43'  8-476498  7060 

1-523502 

8-476693 

7066 

11-523307 

•0001951  9-999805 

17 

44  8-480693 

6991  jl-519307 

8-480892 

6998 

11-519108 

•000199 

9-999801 

16 

45 

8-484848 

6924 

1-515152 

8-485050  6931 

11-514950 

•000203 

9-999797 

15 

46  8-4889631  6859 
47;  8-493040  6794 

1  -511037  J8-489170'  6865 
1-506960  8-493250  6801 

11-510830 
11-506750 

•000206 
•000210 

9-999794 
9-999790 

14 
13 

488-497078!  6731 

1-502922  J8-497293  6738 

11-502707 

•000214 

9-999786 

12 

49;;8-501080!  6669  1-498920. 

8-501298  6676 

11-498702 

•000218 

9-999782 

11 

50  8-505045  6608  1-494956 

8-505267  6615 

11-494733 

•000222 

9-999778 

10 

51  8-5089741  6548  1-491026! 

8-509200  6555 

11-490800 

•000226 

9-999774 

9 

628-612867  6489  1-487133, 

8-513098  6496  11-486902!  -000231 

9-999769 

8 

53  8-516720  G482  1-483274 
548-520551  63751-479449 

8-516961  6439 

8-520790  6382 

11  -483039  :i  -000285 
H  -479210  -000239 

9-999765 

9-999761 

7 
6 

.V>  <s  -524343  6319  1-4751557 

8-524586  6326  11-475414;  -000243 

9-999757 

5 

508-528102  62641-471898 

8-528349  6272  |11  -471(551  -000247 

9-999753 

4 

578-531828  6211  1-468172 

H-53J080  6218  !11-467920;:-000252 

9-999748 

3 

588-535523:  6158  1-41.54477 

8-535779  .6165.11-46422.1  1-000256 

9-999744 

2 

5'.»;8-53918r>!  6106  1-400814 

8-539447  6113  11-400)58 

•000260 

9-999740  j  1 

00  8-512819  6055  1-457181 

8-543084  6062  11-456916  -000265 

9-99973o   0 

Cusinu.            Secant. 

Cotangent.         ;  Tang      t.  o-ecant. 

S'ne. 

88  DEG. 


LOGARITHMIC   SINES,   ETC. 


2  T)EG. 


' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 

my 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

f 

~~o" 

8-542819 

1-457181 

8-543084 

11-456916 

•000265 

9-999735 

60~ 

1 

8-546422 

6004 

1-453578 

8-546691 

6012 

11-453309 

•000269 

07 

9-999731 

59 

2 

8-549995 

5955 

1-450005 

8-5502685962 

11-449732 

•000274 

07 

9-999726 

58 

3 

8-553539 

5906 

1-446461 

8-55381715914 

11-446183 

•000278 

08 

9-999722 

57 

4 

8-557054 

5858 

1-442946 

8-55733615866 

11-442664 

•000283 

08 

9-999717 

56 

5 

8-560540 

5811 

1-439460 

8-560828 

5819 

11-439172 

•000287 

07 

9-999713 

55 

6 

8-563999 

5765 

1-436001 

8-564291 

5773 

11-435709 

•000292 

08 

9-999708 

54 

7 

8-567431 

5719 

1-432569 

8-567727 

5727 

11-432273 

•000296 

07 

9-999704 

53 

8 

8-570836 

5674 

1-429164 

8-571137 

5682 

11-428863 

000301 

08 

9-999699 

52 

9 

8-574214 

5630 

1-425786 

8-574520 

5638 

11-425480 

000306 

08 

9-999694 

51 

10 

8-577566 

5587 

1-422434 

8-5778775595 

11-422123 

000311 

08 

9-999689 

50 

11 

8-580892 

5544 

1-419108 

8-58120815552 

11-418792 

000315 

07 

9-999685 

49 

12 

8-584193 

5502 

1-415807 

8-584514  5510 

11-415486 

000320 

08 

9-999680 

48 

13 

8-587469 

5460 

1-412531 

8-58779515468 

11-412205 

000325 

08 

9-999675 

47 

14 

8-590721 

5419 

1-409279 

8-59105l!5427 

11-408949 

000330 

08 

9-999670 

46 

15 

8-593948 

5379 

1  -406052 

8-594283  5387 

11-405717 

000335 

08 

9-999665 

45 

16 

8-597152 

5339 

1-402848 

8-597492  5347 

11-402508 

•000340 

08 

9-999660 

44 

17 

8-600332 

5300 

1-399668 

8-600677  5308 

11-399323 

•000345 

08 

9-999655 

43 

18 

8-603489 

5261 

i-396511 

8-603839  5270 

11-396161 

•000350 

08 

9  -999650  142 

19 

8-606623 

5223 

1-393377 

8-606978  5232 

11-393022 

•000355 

08 

9-999645  41 

20 

8-6U9734 

5186 

1-390266 

8-610094  5194 

11-389906 

•000360 

08 

9-999640  40 

21 

8-612823 

5149 

1-387177 

8-6131895158 

11-386811 

•000365 

08 

9-999635  39 

22 

8-615891 

5112 

1-384109 

8-616262!5121 

11-383738 

•000371 

10 

9-999629  38 

23 

8-618937 

5076 

1-381063 

8-6193135085 

11-380687 

•000376 

08 

9-999624  '37 

24 

8-621962 

5041 

1-378038 

8-622343  5050 

11-877657 

•000381 

08 

9-999619 

36 

25 

8-624965 

5006 

1-375035 

8-625352  5015 

11-374648 

•000386 

08 

9-999614 

35 

26 

8-627948 

4972 

1  -372052 

8-628340!4981 

11-371660 

•000392 

10 

9-999608 

34 

27 

8-630911 

4938 

1-369089 

8-6313084947 

11-368692 

•000397 

08 

9-999603 

33 

28 

8-63385414904  1-366146 

8-6342564913 

11-365744 

•000403 

10 

9-999597 

32 

29 

8-636776 

4871  1-363224 

8-637184!4880 

11-362816 

•000408 

08 

9-999592 

31 

30 

8-639680 

4839 

1-360320 

8-640093  4848 

11-359907 

•000414 

08 

9-999586 

30 

31 

8-642563  4806 

1-357437 

8-642982 

4816 

11-357018 

•000419 

10 

9-999581 

29 

32 

8-645428 

4775 

1-354572 

8-645853 

4784 

11-354147 

•000425 

10 

9-999575 

28 

33 

8-648274 

4743 

1-351726 

8-648704 

4753 

11-351296 

•000430 

08 

9-999570 

27 

34 

8-651102 

4712 

1-348898 

8-651537 

4722 

11-348463 

•000436 

10 

9-999564 

26 

35 

8-653911 

4682  1-346089 

8-654352 

4691 

11-345648 

•000442 

10 

9-999558 

25 

36 

8-656702 

465211-343298 

8-657149 

4661 

11-342851 

•000447 

08 

9-999553 

24 

37 

8-659475 

4622J1-340525 

8-659928 

4631 

11-340072 

•000453 

10 

9-999547 

23 

38 

8-662230 

45921-337770 

8-662689 

4602 

11-337311 

•000459 

10 

9-999541 

22 

39 

8-664968 

45631-335032 

8-665433 

4573 

11-334567 

•000465 

10 

9-999535 

21 

40 

8-667689 

46861-882311 

8-668160 

4544 

11-331840 

•000471 

10 

9-999529 

20 

41 

8-670393 

450611-329607 

8-670870 

4516 

11-329130 

•000476 

08 

9-999524 

19 

42 

8-673080 

4479  1-326920 

8-673563 

4488 

11-326437 

•000482 

10 

9-999518 

18 

43 

8-675751 

4451 

1-324249 

8-676239 

4461 

11-323761 

•000488 

10 

9-999512 

17 

44 

8-678405 

4424 

1-321595 

8-678900 

4434 

11-321100 

•000494 

10 

9-999506 

16 

45 

8-681043 

4397 

1-318957 

8-681544 

4407 

11-318456 

•000500 

10 

9-999500 

15 

46 

8-683665 

4370 

1-316335 

1  8-684172 

4380 

11-315828 

•000507 

10 

9-999493 

14 

47 

"8-686272 

4344 

1-313728 

8-686784 

4354 

11-313216 

•000513 

12 

9-999487 

13 

48 

8-688863 

4318 

1-311137 

•  8-689381 

4328 

11-310619 

•000519 

10 

9-999481 

12 

49 

8-691438 

4292  i  1-308562 

!  8-691963 

4303 

11-308037 

•000525 

10 

9-999475 

11 

50 

8-693998 

4267|l  -306002 

8-694529 

4277 

11-305471 

•000531 

10 

9-999469 

10 

51 

8-696543 

4242 

1-303457 

8-697081 

4252 

11-302919 

•000537 

12 

9-999463 

9 

52 

8-699073 

4217 

1-300927  8-699617 

4228 

11-300383 

•000544 

10 

9-999456 

8 

53 

8-701589 

4192 

1-298411  i  8-702139 

4203 

11-297861 

•000550 

10 

9-999450 

7 

54 

8-704090 

416811-295910  !  8-704646 

4179 

11-295354 

•000557 

10 

9-999443 

6 

55 

8-706577 

41441-293423  8-707140 

4155 

11-292860 

•000563 

12 

9-999437 

5 

56 

8-709049 

41211-290951  8-709618 

4132 

11-290382 

•000569 

10 

9-999431 

4 

57 

8-711507 

40971-288493  8-712083 

4108 

11-287917 

•000576 

12 

9-999424 

3 

58 

8-713952 

40741-286048  '8-714534 

4085 

11-285466 

•000582 

10 

9-999418 

2 

59 

8-716383 

4052  1-283617  ;  8-716972 

4062 

11-283028 

•000589 

12 

9-999411 

1 

60 

8-718800 

40291-281200  8-719396 

4040 

11-280604 

•000596 

12 

9-999404 

£ 

' 

Cosine. 

Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

87  BEG. 


44 


LOGARITHMIC    SINES,    ETC. 


3  DE(J. 


1 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 

mo'- 

Cotangent. 

Secant. 

SJ'j  Cosine. 

' 

0 

8-718800 

1-281200 

8-719396 

11-280604 

•000596 

19-999404 

60 

1 

8-721204 

4006 

1-278796 

8-721806 

4017 

11-278194 

•000602 

10  9-999398 

59 

2 

8-723595 

3984 

1-276405 

8-724204 

3995 

11-275796 

•000609 

12i9-999391 

58 

3 

8-725972 

3962 

1-274028 

8-726588 

3974 

11-273412 

•000616 

12  9-999384 

57 

4 

8-728337 

3941 

1-271663 

8-728959 

3952 

11-271041 

•000622 

10  9-999378 

56 

6 

8-730688 

3919 

1-269312 

8-731317 

3931 

11-268683 

•000629 

12  .9-999371 

55 

6 

8-733027 

3898 

1-266973 

8-733663 

3909 

11-266337 

•000636 

12  9-999364 

54 

7 

8-735354 

3877 

1-264646 

8-735996 

3889J11-264004 

•000643 

12  9-999357 

53 

8 

8-737667 

3857 

1-262333 

8-738317 

3868 

11-261683 

•000650 

12  9-999350 

52 

9 

8-739969 

3836 

1-260031 

8-740626 

3848 

11-259374 

•000657 

12  9-999343 

51 

10 

8-742259 

3816 

1-257741 

8-742922 

3827 

11-257078 

•000664 

12  9-999336 

50 

11 

8-744536 

3796 

1-255464 

8-745207 

3807 

11-254793 

•000671 

12  9-999329 

49 

12 

8-746802 

3776 

1-253198 

8-747479 

3787 

11-252521 

•000678 

12  9-999322 

48 

13 

8-749055 

3756 

1-250945 

8-749740 

3768 

11-250260 

•000685 

12  9-999315 

47 

14 

8-751297 

3737 

1-248703 

8-751989 

3749 

11-248011 

•000692 

12  9-999308 

46 

15 

8-753528 

3717 

1-246472 

8-754227 

3729 

11-245773 

•000699 

12  9-999301 

45 

16 

8-755747 

3698 

1-244253 

8-756453 

3710 

11-243547 

•000706 

12  9-999294 

44 

17 

8-757955 

3679 

1-242045 

8-758668 

3692 

11-241332 

•000713 

13  9-999287 

43 

18 

8-760151 

3661 

1-239849 

8-760872 

3673 

11-239128 

•000721 

12  9-999279 

42 

19 

8-762337 

3642 

1-237663 

8-763065 

3655 

11-236935 

•000728 

12  9-999272 

41 

20 

8-764511 

3624 

1-235489 

8-765246 

3636 

11-234754 

•000735 

12  9-999265 

40 

21 

8-766675 

3606 

1-233325 

8-767417 

3618 

11-232583 

•000743 

13  9-999257 

39 

22 

8-768828 

3588 

1-231172 

8-769578 

3600 

11-230422 

•000750 

12  9-999250 

38 

23 

8-770970 

3570 

1-229030 

8-771727 

3583 

11-228273 

•000758 

13  9-999242 

37 

24 

8-773101 

3553 

1-226899 

8-773866 

3565 

11-226134 

•000765 

12  9-999235 

36 

25 

8-775223 

3535 

1-224777 

8-775995 

3548 

11-224005 

•000773 

13  9-999227 

35 

26 

8-777333 

3518 

1-222667 

8-778114 

3531 

11-221886 

•000780 

12  9-999220 

34 

27 

8-779434 

3501 

1-220566 

8-780222 

3514 

11-219778 

•000788 

13  9-999212 

33 

28 

8-781524 

3484 

1-218476 

8-782320 

3497 

11-217680 

•000795 

13  9-999205 

32 

29 

8-783605 

3467 

1-216395 

8-784408 

3480 

11-215592 

•000803 

13  9-999197 

31 

30 

8-785675 

3451 

1-214325 

8-786486 

3464 

11-213514 

•000811 

13  9-999189 

30 

31 

8-787736 

3434 

1-212264 

8-788554 

3447 

11-211446 

•000819 

13  9-999181 

29 

32 

8-789787 

3418 

1-210213 

8-7906133431 

11-209387 

•000826 

12 

9-999174 

28 

33 

8-791828 

3402 

1-208172 

8-792662 

3415 

11-207338 

•000834 

13 

9-999166 

27 

34 

8-793859 

3386 

1-206141 

8-794701 

3399 

11-205299 

•000842 

13 

9-999158 

26 

35 

8-795881 

3370 

1-204119 

8-796731 

3383 

11-203269 

•000850 

13 

9-999150 

25 

36 

8-797894 

3354 

1-202106 

8-798752 

3368 

11-201248 

•000858 

13 

9-999142 

24 

37 

8-799897 

3339 

1-200103 

8-800763 

3352 

11-199237 

•000866 

13 

9-999134 

23 

38 

8-801892 

3323 

1-198108 

8-802765 

3337 

11-197235 

•000874 

13 

9-999126 

22 

39 

8-803876 

3308 

1-196124 

8-804758 

3322 

11-195242 

•000882 

13 

9-999118 

21 

40 

8-805852 

3293 

1-194148 

8-806742 

3306 

11-193258 

•000890 

13 

9-999110  20 

41 

8-807819 

3278 

1-192181 

8-808717 

3292 

11-191283 

•000898 

13 

9-999102  19 

42 

8-809777 

3263 

1-190223 

8-810683 

3277 

11-189317 

•000906 

13 

9-99909418 

43 

8-811726 

3249 

1-188274 

8-812641 

3262 

11-187359 

•000914 

13 

9-999086  17 

44 

8-813667 

3234 

1-186333 

8-814589 

3248 

11-185411 

•000923 

15 

9-999077  16 

45 

8-815599 

3219 

1-184401 

8-816529 

3233 

11-183471 

•000931 

13 

9-999069 

15 

46 

8-817522 

3205 

•182478 

8-818461 

3219 

11-181539 

•000939 

13 

9-999061 

14 

47 

8-819436 

3191 

•180564 

8-820384 

3205 

11-179616 

•000947 

15 

9-999053 

13 

48 

8-821343 

3177 

•178657 

8-822298 

3191 

11-177702 

•000956 

13 

9-999044 

12 

49 

8-823240 

3163 

•176760 

8-82T4205 

3177 

11-175795 

•000964 

13 

9-999036 

11 

50 

8-825130 

3149 

•174870 

8-826103 

3163 

11-173897 

•000973 

15 

9-999027 

10 

51 

8-827011 

3135 

•172989 

8-827992 

3150 

11-172008 

•000981 

13 

9-999019 

9 

62 

8-828884 

3122 

•171116 

8-829874 

3136 

11-170126 

•000990 

15 

9-999010 

8 

53 

8-830749 

3108 

•169251 

8-831748 

3123 

11-168252 

•000998 

13 

9-999002 

7 

54 

8-832607 

3095 

•167393 

8-833613 

3109 

11-166387 

•001007 

15 

9-998993 

6 

55 

8-834456 

3082 

•165544 

8-835471 

3096 

11-164529 

•001016 

15 

9-998984 

5 

56 

8-836297 

3069 

•163703 

8-837321 

3083 

11-162679 

•001024 

13 

9-998976 

4 

57 

8-838130 

3056 

•161870 

8-839163 

3070 

11-160837 

•001033 

15 

9-998967 

3 

58 

8-839956 

3043 

•160044 

8-840998 

3057 

11-159002 

•001042  15 

9-998958 

2 

59 

8-841774 

3030 

1-158226 

8-842825 

3045 

11-157175 

•001050  13 

9-998950 

1 

60 

8-843585 

3017 

1-156415 

8-844644 

3032 

11-155356 

•001059  15 

9-998941 

0 

' 

Cosine. 

i   Secant. 

Cotangent. 

Tangent.   '  Cosecant.  \ 

Sine. 

' 

86  DKG. 


LOGARITHMIC    SINES,    ETC. 


45 


4  DBG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 

100" 

Cosine. 

' 

0 

8-843585 

1-156415 

8-844644 

1-155356 

001059 

9-998941 

60 

1 

8-845387 

3005 

1-154613 

8-846455 

3019 

11-153545 

001068 

15 

9-998932 

59 

2 

8-847183 

2992 

1-152817 

8-848260 

3007 

11-151740 

001077 

15  9-998923 

58 

3 

8-848971 

2980 

1-151029 

8-850057 

2995 

11-149943 

001086 

15  9-998914 

57 

4 

8-850751 

2967 

1-149249 

8-851846 

2982 

11-148154 

001095 

15 

9-998905 

56 

5 

8-852525 

2955 

1-147475 

8-853628 

2970 

11-146372 

001104 

15 

9-998896 

55 

6 

8-854291 

2943 

1-145709 

8-855403 

2958 

11-144597 

001113 

15)9-998887 

54 

7 

8-856049 

2931 

1-143951 

8-857171 

2946 

11-142829 

001122 

15 

9-998878 

53 

8 

8-857801 

2919 

1-142199 

8-858932 

2935 

11-141068 

001131 

15 

9-998869 

52 

9 

8-859546 

2908 

1-140454 

8-860686 

2923 

11-139314 

001140 

15 

9-998860 

51 

10 

8-861283 

2896 

1-138717 

8-862433 

2911 

11-137567 

001149 

15 

9-998851 

50 

11 

8-863014 

2884 

1-136986 

8-864173 

2900 

11-135827 

001159 

17 

9-998841 

49 

12 

8-864738 

2873 

1-135262 

8-865906 

2888 

11-134094 

001168 

15 

9-998832 

48 

13 

8-866455 

2861 

1-133545 

8-867632 

2877 

11-132368 

001177 

15 

9-998823 

47 

14 

8-868165 

2850 

1-131835 

8-869351 

2866 

.1-130649 

001187 

17 

9-998813 

46 

15 

8-869868 

2839 

1-130132 

8-871064 

2854 

11-128936 

001196 

15 

9-998804 

45 

16 

8-871565 

2828 

1-128435 

8-872770 

2843 

11-127230 

001205 

15 

9-998795 

44 

17 

8-873255 

2817 

1-126745 

8-874469 

2832 

11-125531 

001215 

17 

9-998785 

43 

18 

8-874938 

2806 

1-125062 

8-876162 

2821 

11-123838 

001224 

15 

9-998776 

42 

19 

8-876615 

2795 

1-123385 

8-877849 

2811 

11-122151 

001234 

17 

9-998766 

41 

20 

8-878285 

2784 

1-121715 

8-879529 

2800 

11-120471 

001243 

15 

9-998757 

40 

21 

8-879949 

2773 

1-120051 

8-881202 

2789 

11-118798 

001253 

17 

9-998747 

39 

22 

8-881607 

2763 

1-118393 

8-882869 

2779 

11-117131 

001262 

17 

9-998738 

38 

23 

8-883258 

2752 

1-116742 

8-884530 

2768 

11-115470 

•001272 

15 

9-998728 

37 

24 

8-884903 

2742 

1-115097 

8-886185 

2758 

11-113815 

•001282 

17 

9-998718 

36 

25 

8-886542 

2731 

1-113458 

8-887833 

2747 

11-112167 

•001292 

17 

9-998708 

35 

26 

8-888174 

2721 

1-111826 

8-889476 

2737 

11-110524 

•001301 

15 

9-998699 

34 

27 

8-889801 

2711 

1-110199 

8-891112 

2727 

11-108888 

•001311 

17 

9-998689 

33 

28 

8-891421 

2700 

1-108579 

8-892742  2717 

11-107258 

•001321 

17 

9-998679 

32 

29 

8  -898035 

2690 

1-106965 

8-8943662707 

11-105634 

•001332 

17 

9-998669 

31 

30 

8-894643 

2680 

1-105357 

8-895984  2697 

11-104016 

•001341 

17 

9-998659 

30 

31 

8-896246 

2670 

1-103754 

8-897596 

2687 

11-102404 

•001351 

17 

9-998649 

%> 

32 

8-897842 

2660 

1-102158 

8-899203 

2677 

11-100797 

-001361 

17 

9-998639 

28 

33 

8-899432 

2651 

1-100568 

8-900803 

2667 

11-099197 

•001371 

17 

9-998629 

27 

34 

8-901017 

2G41 

1-098983 

8-902398 

2658 

11-097602 

•001381 

17 

9-998619 

26 

35 

8-902596 

2631 

1-097404 

8-903987 

2648 

11-096013 

•001391 

17 

9-998609 

25 

36 

8-904169 

2622 

1-095831 

8-905570 

2638 

11-094430 

•001401 

17 

9-998599 

24 

37 

8-905736 

2612 

1-094264 

8-907147 

2629 

11-092853 

•001411 

17 

9-998589 

23 

38 

8-907297 

2603 

1-092703 

8-908719 

2620 

11-091281 

•001422 

18 

9-998578 

22 

39 

8-908853 

2593 

1-091147 

8-910285 

2610 

11-089715 

•001432 

17 

9-998568 

21 

40 

8-910404 

2584 

1-089596 

'8-911846 

2601 

11-088154 

•001442 

17 

9-998558 

20 

41 

8-911949 

2575 

1-088051 

1  8-913401 

2592 

11-086599 

•001452 

17 

9-998548 

19 

42 

8-913488 

2566 

1-086512 

I  8-914951 

5283 

11-085049 

•001463 

18 

9-998537 

18 

43 

8-915022 

2556 

1-084978 

8-916495 

2574 

11-083505 

•001473 

17 

9-998527 

17 

44 

8-916550 

2547 

1-083450 

8-918034 

2565 

11-081966 

•001484 

18 

9-998516 

16 

45 

8-918073 

2538 

1-081927 

8-919568 

2556 

11-080432 

•001494 

17 

9-998506 

15 

46 

8-919591 

2529 

1-080409 

i  8-921096 

2547 

11-078904 

•001505 

18 

9-998495 

H 

47 

8-921103 

2520 

1-078897 

8-922619 

2538 

11-077381 

•001515 

17 

9-998485 

13 

48 

8-922610 

2512 

1-077390 

8-924136 

2530 

11-075864 

•001526 

18 

9-998474 

12 

49 

8-924112 

2503 

1-075888  8-925649 

2521 

11-074351 

•001536 

17 

9-998464 

11 

50 

8-925001 

2494 

1-074391  J8-927156 

2512 

11-072844 

•001547 

18 

9-998453 

10 

51 

8-927100 

248b 

1-072900  '8-928658 

2503 

11-071342 

•001558 

18 

9-998442 

9 

52 

8-928587 

2477|l-071413  8-930155 

2495 

11-069845 

•001569 

18 

9-998431 

8 

53 

8-930068 

2469  1-069932  8-931647 

2486 

11-068353 

•001579 

17 

9-998421 

7 

54 

8-931544 

2460 

1-068456  8-933134 

2478 

11-066866 

•001590 

18 

9-998410 

6 

55 

8-933015 

2452 

1  -066985  ;  8-934616 

2470 

11-065384 

•001601 

18 

9-998399 

5 

56 

8-934481 

2443 

1-065519  8-936093 

2461 

11-063907 

•001612 

18 

9-998388 

4 

57 

8-935942 

2435 

1-064058  8-937565 

2453 

11-062435 

•001623 

18 

9-998377 

3 

58 

8-937398 

2427 

1-062602  8-939032 

2445 

11-060968 

'•001634 

18 

9-998366 

2 

59 

8-938850 

2419 

1-061150  8-9404942437 

11-059506 

•001645 

18 

9-998355 

1 

60 

8-94029t 

2411 

1-059704  8-9419522429 

11-058048 

•001656 

18 

9-998344 

0 

' 

Cosine. 

!   Secaut.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

85  DEO. 


46 


LOGARITHMIC    SINES,    ETC 


5  DEG. 


• 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

IMff. 
100" 

Cotangent.    Secant. 

Dirt'. 
100" 

Cosine. 

! 

~~0 

8-940296 

1-059704 

8-941952 

11-058048  1-001656 

JMJ98344 

60~ 

1 

8-941738 

2403 

1-058262 

8-943404 

2421 

11-056596  -001667 

18 

9-998333 

59 

2 

8-943174 

2394 

1-056826 

8-944852 

2413 

11-055148  :  -001678 

18 

9-998322 

58 

3 

8-944606 

2387 

1-055394 

8-946295 

2405 

11-053705  1-001689 

18 

9  -998311!  57  ! 

4 

8-946034 

2379 

1-053966 

8-947734 

2397 

11-052266 

•001700 

18 

9-998800  156 

5 

8-947456 

2371 

1-052544 

8-949168 

2390 

11-050832 

•001711 

18 

9-998289 

55 

6 

8-948874 

2363 

1-051126 

8-950597 

2382 

11-049403 

•001723 

20 

9-998277 

54 

7 

8-950287 

2355 

1-049713 

8-952021 

2374 

11-047979 

•001734 

18 

9-998266 

53 

8 

8-951696 

2348 

1-048304 

8-953441 

2366 

11-046559 

•001745 

18 

9-998255 

52 

9 

8-953100 

2340 

•046900 

8-954856 

2359 

11-045144 

•001757 

20 

9-998243 

51 

10 
11 

8-954499 
8-955894 

2332 
2325 

•045501 
•044106 

8-956267 
8-957674 

2351 
2344 

11  -0437331:  -001768 
11-042326  -001780 

18 
20 

9-998232 
9-998220 

50 
49 

12 

8-957284 

2317 

•042716 

8-959075 

2336 

11-040925  -001791 

18 

9-99820'.) 

48 

13 

8-958670 

2310 

•041330 

8-960473 

2329 

11-039527  -001803 

20 

9-998197 

47 

14 

8-960052 

2302 

•039948 

8-961866 

2322 

11-038134 

•001814 

18 

9-998186 

46 

15 

8-961429 

2295 

•038571 

8-963255 

2314 

11-036745 

•001826 

20 

9-998174^45 

16 

8-962801 

2288 

•037199 

8-964639 

2307 

11-035361 

•001837 

18 

9-998163'  44 

17 

8-964170 

2280 

•035830 

8-966019 

2300 

11-033981 

•001849 

20 

9-998151  43 

18 

8-965534 

2273 

•034466 

8-967394 

2293 

11-032606 

•001861 

20 

9-998139:42 

19 

8-966893 

2266 

•033107 

8-968766 

2286 

11-031284 

•001872 

18 

9-998128 

41 

20 

8-968249 

2259 

1-031751 

8-970133 

2279 

11-029867 

•001884 

20 

9-998116 

40 

21 

8-969600 

2252 

1-030400 

8-971496 

2271 

11-028504 

•001896 

20 

9-998104 

39 

22 

8-970947 

2245 

1-029053 

8-972855 

2265 

11-027145 

•001908 

'20 

9-998Q92  38 

23 

8-972289 

2238  1-027711 

8-974209 

2257 

11-025791 

•001920 

20 

9-998080  i37 

24 

8-973628 

2231  1-026372 

8-975560 

2251 

11-024440  -001932 

20 

9-998068i36 

25 

26 

8-974962 
8-976293 

22241-025038 
2217  1-023707 

8-976906 
8-978248 

2244 
2237 

11-023094 
11-021752 

•001944 
•001956 

20 
20 

9-998056  35 
9-998044  34 

27 
28 

8-977619 
8-978941 

22101-022381 
2203  1-021059 

8-979586 
8-980921 

2230  11-020414 
2223  11-019079 

•001968 
•001980 

20 
20 

9-998032  33 
9-998020  :-)2 

29 

8-980259 

2197 

1-019741 

8-982251 

2217 

11-017749 

•001992 

20 

9-998008  ::i 

30 

8-981573 

2190 

1-018427 

8-983577 

2210 

11-016423 

•002004 

20 

9-997996,30 

31 

8-982883 

2183 

1-017117 

8-984899 

2204 

11-015101 

•002016 

20 

9-997984;  29 

32 

8-984189 

2177 

1-015811 

8-986217 

2197 

11-013783 

•002028 

20 

9-997972  28 

33 

8-985491 

2170 

1-014509 

8-987532 

2191 

11-012468 

•002041 

22 

9-997951)  27 

34 

8-986789 

2163 

1-013211 

8-988842 

2184 

11-011158 

•002053 

20 

9-997947 

26 

35 

8-988083 

2157 

1-011917 

8-990149 

2178 

11-009851 

•002065 

20 

9-997935 

25 

36 

8-989374 

2150 

1-010626 

8-991451 

2171 

11-008549 

•002078 

22 

9-997922  24 

37 

8-990660 

2144 

1-009340 

8-992750 

2165 

11-007250 

•002090 

20 

9-997910 

•2:>> 

38 

8-991943 

2138 

1-008057 

8-994045 

2158 

11-005955 

•002103 

22 

9-997897 

22 

39 

8-993222 

2131 

1-006778 

8-995337 

2152 

11-004663 

•002115 

20 

9-997885 

21 

40 

8-994497 

2125 

1-005503 

8-996624 

2146 

11-003376 

•002128 

-22 

9-997872 

20 

41 

8-995768 

2119 

1-004232 

8-997908 

2140 

11-002092 

•002140 

20 

9-997860 

19 

42 

8-997036 

2112 

1-002964 

8-999188 

2134 

11-000812 

•002153 

22 

9-997847 

18 

43 

8-998299 

2106 

1-001701 

9-000465 

2127 

10-999535 

•002165 

20 

9-997835 

17 

44 

8-999560 

2100 

1-000440 

9-001738 

2121 

10-998262 

•002178 

22 

9-997822 

16 

45 

9-000816 

2094 

0-999184 

9-003007 

2115 

10-996993 

•002191 

22 

9-997809 

15 

46 
47 

9-002069 
9-003318 

2088 
2082 

0-997931 
0-996682 

9-004272 
9-005534 

2109 
2103 

10-995728 
10-994466 

•002203 
•002216 

20 

22 

9-997797 

9-997784 

14 
13 

48 

9-004563 

2076 

0-995437 

9-006792 

2097 

10-993208 

•002229 

22 

9-997771 

12 

49 

9-005805 

2070 

0-994195 

9-008047 

2091 

10-991953 

•002242 

22 

9-997758 

11 

50 

9-007044 

2064 

0-992956 

9-009298 

2085 

10-990702 

•002255 

22 

9-997745 

10 

51 

9-008278 

2058 

0-991722 

9-010546 

2079 

10-989454 

•002268 

22 

9-997732 

9 

52 

9-009510 

2052 

0-990490 

9-011790 

2074 

10-988210 

•002281 

22 

9-997719 

8 

53 

9-010737 

2046 

0-989263 

9-013031 

2068 

10-986969 

•002294 

22 

9-997706 

7 

54 

9-011962 

2040 

0-988038 

9-014268 

2062 

10-985732 

•002307 

22 

9-997693 

6 

55 

9-013182 

2034 

0-986818 

9-015502 

2056 

10-984498 

•002320 

22 

9-997680 

5 

56 

9-014400 

2029 

0-985600 

9-016732 

2051 

10-983268 

•002333 

22 

9-997667 

4 

57 

9-015613 

2023 

0-984387 

9-017959 

2045 

10-982041 

•002346 

22 

9-997654 

3 

58 

9-016824 

2017 

0-983176 

9-019183 

2039 

10-980817 

•002359 

22 

9-99764H 

2 

59 

9-018031 

2012 

0-981969 

9-020403 

2034 

10-979597 

•002372 

22 

9-997628, 

1 

60 

9-019235 

2006 

0-980765 

9-021620 

2028 

10-978380 

•002386 

23 

9-997614! 

0 

' 

Cosine. 

'   Secant,   i  Cotangent. 

Tangent. 

Cosecant. 

Sine. 

84-  BEG. 


LOGARITHMIC    SINES,  ETC. 


47 


6  DEG. 


1  , 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 

100" 

j  Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

' 

~~0~ 

9  -019235 

•980765 

9-021620 

10-978380 

•002386 

9-997614 

60~ 

1 

9-020435 

2000 

•979565 

9-022834 

2023 

10-977166 

•002399 

22 

9-997601 

59 

2 

9-021632 

1995 

•978368 

9-024044 

2017 

10-975956 

•002412 

22 

9-997588 

58 

3 

9-022825 

1989 

•977175 

9-025251 

2011 

10-974749 

•002426 

23 

9-997574 

57 

4 

9-024016 

1984 

•975984 

9-026455 

2006 

10-973545 

•002439 

22 

9-997561  56 

5 

9-025203 

1978 

•974797 

9-027655 

2001 

10-972345 

-002453 

23 

9-997547  £5 

6 

9-026386 

1973 

•973614 

9-028852 

1995 

10-971148 

•002466 

22 

9-997534  54 

7 

9-027567 

1967 

•972433 

9-030046 

1990 

10-969954 

•002480 

23 

9-997520  53 

8 

9-028744 

1962 

•971256 

9-031237 

1985 

10-968763 

•002493 

22 

9-997507  52 

9 

9-029918 

1957 

•970082 

9-032425 

1979 

10-967575 

•002507 

23 

9-997493  51 

10 

9-031089 

1951 

•968911 

9-033609 

1974J10-966391 

•002520 

22 

9-997480'  50 

11 

9-032257 

1946 

•967743 

9-034791 

1969 

10-965209 

•002534 

23 

9-997466  49 

12 

9-033421 

1941 

•966579 

9-035969 

1964 

10-964031 

•002548 

23 

9-997452  48 

13 

9-034582 

1936 

•965418 

9-037144 

1958 

10-962856 

•002561 

22 

9-997489  47 

14 

9-035741 

1930 

•964259 

9-038316 

1953;iO-961684 

•002575 

23 

9-997425|46 

15 

9-036896 

1925 

•963104 

9-039485 

1948 

10-960515 

•002589  23 

9-997411  45 

16 

9-038048 

1920 

•961952 

9-040651 

1943 

10-959349 

•002603  23 

9-997397;  44 

17 

9-039197 

1915 

•960803 

9-041813 

1938  10-958187 

•002617 

23 

9-997383  43 

18 

9-040342 

1910 

•959658 

9-042973 

1933 

10-957027 

•002631 

23 

9  -997369  '42 

19 

9-041485 

1905 

•958515 

9-044130 

1928 

10-955870 

•002645 

23 

9-997355:41 

20 

9-042625 

1899 

•957375 

9-045284 

1923 

10-954716 

•002659 

23 

9-99734L40 

21 

9-043762 

1895 

•956238 

9-046434 

1918  10-953566 

•002673 

23 

9-997327  39 

22 

9-044895 

1889 

•955105 

9-047582 

1913 

10-952418 

•002687 

23 

9-997313  38 

23 

9-046026 

1884 

•953974 

9-048727 

190810-951273 

•002701 

23 

9-997299  37 

24 

9-047154 

1879 

•952846 

9-049869 

1903J10-950131 

•002715 

23 

9-997285  36 

25 

9-048279 

1875 

•951721 

9-051008 

1898 

10-948992 

•002729 

23 

9-997271l35 

26 

9-049400 

1870 

•950600 

9-052144 

1893 

10-947856 

•002743 

23 

9.997257  34 

27 

9-050519 

1865 

•949481 

9-053277 

1889 

10-946723 

•002758 

25 

9-997242  33 

28 

9-051635 

1860 

•948365 

9-054407 

1884 

10-945593 

•002772 

23 

9-997228;  32 

29 

9-052749 

1855 

•947251 

9-055535 

1879llO-9l4465 

•002786 

23 

9-997214:31 

30 

9-053859 

1850 

•946141 

9-056659 

1874ilO-943341 

•002801 

25 

9-997199 

30 

31 

9-054966 

1845 

•945034 

9-057781 

1870 

10-942219 

•002815 

23 

9-997185 

29 

32 

9-056071 

1841 

•943929 

9-058900 

1865 

10-941100 

•002830 

25 

9-997170 

28 

33 

9-057172 

1836 

•942828 

9-060016  1860 

10-939984 

•002844 

23 

9-997156 

27 

34 

9-058271 

1831 

•941729 

9-061130  1855 

10-938870 

•002859 

25 

9-997141 

26 

35 

9-059367 

1827 

•940633 

9-062240il85lllO-937760 

•002873 

23 

9-997127 

25 

36 

9-060460 

1822 

•939540 

9-063348 

1846  10-936652 

•002888 

25 

9-997112 

24 

37 

9-061551 

1817 

•938449 

'  9  -064453 

1842(10-935547 

•002902 

23 

9-997098 

23 

38 

9-062639 

1813 

•937361 

9-065556 

1837 

10-934444 

•002917 

25 

9-997083 

22 

39 

9-063724 

1808 

•936276 

9-066655 

1833!lO-933345 

•002932 

25 

9-997068 

21 

40 

9-064806 

1804 

•935194 

9-067752 

1828 

10-932248 

•002947 

25 

9-997053 

20 

41 

9-065885 

1799 

•934115 

9-068846 

1824 

10-931154 

•002961 

23 

9-997039 

19 

42 

9-066962 

1794 

•933038  19-069938 

1819 

10-930062 

•002976 

25 

9-997024 

18 

43 

9-068036 

1790 

•931964  9-071027 

1815110-928973 

•002991 

25 

9-997009 

17 

44 

9-069107 

1786 

•930893 

9-072113 

1810 

10-927887 

•003006 

25 

9-996994 

16 

45 

9-070176 

1781 

•929824 

9-073197 

1806J10-926803 

•003021 

25 

9-996979 

15 

46 

9-071242 

1777 

•928758 

9-074278 

1802 

10-925722 

•003036 

25 

9-996964 

14 

1  47 

9-072306 

1772 

•927694 

9-075356 

1797 

10-024644 

•003051 

25 

9-996949 

U 

48 

9-073366 

1768 

•926634 

9-076432 

1793 

10-923568 

•003066 

25 

9-996934 

12 

49 

9-074424 

1763 

•925576 

9-077505 

1789 

10-922495 

•003081 

25 

9-996919 

11 

50 

9-075480 

1759 

•924520 

9.078576  1784 

10-921424 

•003096 

25 

9-996904 

10 

51 

9-076533 

1755 

•923467 

9-079644  1780 

10-920356 

•003111 

25 

9-996889 

9 

52 

9-077583 

1750  -922417 

9-080710  1776 

10-919290 

•003126 

25 

9-996874 

8 

53 

9-078631 

17461  -921369 

9-081773  1772 

10-918227 

•003142 

27 

9-996858 

7 

54 

9-079676 

1742 

•920324 

9-082833  1767 

10-917167 

•003157 

25 

9-996843 

6 

55 

9-080719 

1738 

•919281 

9-083891 

1763 

10-916109 

•003172 

25 

9-996828 

5 

56 

9-081759 

1733 

•918241 

9-084947 

1759 

10915053 

•003188  27 

9-996812 

4 

57 

9-082797 

1729 

•917203 

9-086000  1755 

10-914000 

•0032031  25 

9-996797 

3 

58 

9-083232 

1725 

•916168 

9-087050 

1751 

10-912950 

•003218  25 

9-996782 

2 

59 

9-084864 

1721 

•915136 

9-088098  1747 

10-911902  '-003234 

27 

9-996766 

1 

60 

9-085894 

•914106 

9-089144  1743 

10-910856  -003249 

25 

9-996751 

0 

/ 

Cosine. 

Secant,   ij  Cotangent,  i 

Tangent.  !  Cosecant.  | 

Sine. 

i  i 

83  DEG. 


48 


LOGARITHMIC    SINES,  ETC. 


7  DEG. 


• 

Sine. 

Diff. 
100" 

Cosecant,  i  Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

i' 

0 

9-085894 

•914106  i  9-089144 

10-910856 

•003249 

9-996751 

60" 

1 

9-086922 

1713 

•913078  1  9-090187 

1738 

10-909813 

•003265 

27 

9-996735 

59 

2 

9-087947 

1709 

•912053 

9-091228 

1735 

10-908772 

•003280 

25 

9-996720  58 

3 

9-088970 

1704 

•911030 

9-092266 

1731 

10-907734 

•003296 

27 

9-996704157 

4 

9-089990 

1700 

•910010 

9-093302 

1727 

10-906698 

•003312 

27 

9-996688  156 

5 

6 

9-091008 
9-092024 

1696 
1692 

•908992 
•907976 

9-094336 
9-095367 

1722 
1719 

10-905664 
10-904633 

•003327 
•003343 

25 

27 

9-996673  55 
9-996657  54 

7 

9-093037 

1688 

•906963 

9-096395 

1715 

10-903605 

•003359 

27 

9-996641 

53 

8 

9-094047 

1684 

•905953 

9-097422 

1711 

10-902578 

•003375 

27 

9-996625 

52 

9 

9-095056 

1680 

•904944 

9-098446 

1707 

10-901554 

•003390 

25 

9-996610 

51 

10 

9-096062 

1676 

•903938 

9-099468 

1703 

10-900532 

•003406 

27 

9-996594 

50 

11 

9-097065 

1673 

•902935 

9-100487 

1699 

10-899513 

•003422 

27 

9-996578 

49 

12 

9-098066 

1668 

•901934 

9-101504 

1695 

10-898496 

•003438 

27 

9-996562 

48 

13 

9-099065 

1665 

•900935 

.9-102519 

1691 

10-897481 

003454 

27 

9-996546 

47 

14 

9-100062 

1661 

•899938 

9-103532 

1687 

10-896468 

•003470 

27 

9-996530 

46 

15 

9-101056 

1657 

•898944 

9-104542 

1684 

10-895458 

•003486 

27 

9-996514J45 

16 

9-102048 

1653 

•897952 

9-105550 

1680 

10-894450 

•003502 

27 

9-996498H44 

17 

9-103037 

1649 

•896963 

9-106556 

1676 

10-893444 

•003518 

27 

9-996482II43 

18 

9-104025 

1645 

•895975 

9-107559 

1672 

10-892441 

•003535 

28 

9-996465142 

19 

9-105010 

1642 

•894990 

9-108560 

1669 

10-891440 

•003551 

27 

9-996449|41 

20 

9-105992 

1638 

•894008 

9-109559 

1665 

10-890441 

•003567 

27 

9-996433  40 

21 

9-106973 

1634 

•893027 

9-110556 

1661 

10-889444 

•003583 

27 

9-996417139 

22 

9-107951 

1630 

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9-111551 

1658 

10-888449 

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28 

9-996400J  38 

23 

9-108927 

1627 

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9-112543 

1654 

10-887457 

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27 

9-996384137 

24 

9-109901 

1623 

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9-113533 

1650 

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27 

9-996368J36 

25 

9-110873 

1619 

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9-114521 

1647 

10-885479 

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28 

9-996351135 

26 

9-111842 

1616 

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9-115507 

1643 

10-884493 

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27 

9-996335 

34 

27 

9-112809 

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1639 

10-883509 

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28 

9-996318 

33 

28 

9-113774 

1608 

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9-117472 

1636 

10-882528 

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9-996302 

32 

29 

9-114737 

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31 

30 

9-115698 

1601 

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32 

9-117613 

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9-996235 

28 

33 

9-118567 

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9-996219 

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34 

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9-123317 

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9-996202 

26 

35 

9-120469 

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9-124284 

1611  10-875716 

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9-996185 

25 

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9-121417 

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9-996168 

24 

37 

9-122362 

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9-996151 

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9-123306  1573 

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9-125187 

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9-126125 

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9-990083 

19 

42 

9-127060 

1559 

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9-130994 

1587  10-869006 

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28 

9-996066 

18 

43 

9-127993 

1556 

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9-131944 

1584  10-868056 

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28 

9-996049 

17 

44 

9-128925  1552 

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9-132893 

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9-996032 

16 

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9-129854ll549 

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9-996015 

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9-130781  1545 

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9-134784 

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9-995998 

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9-131706|1542 

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9-135726 

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10-864,274 

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30 

9-995980 

13 

48 

9-132630  1639 

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9-136667 

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10-863333 

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28 

9-995963 

12 

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9-133551  1535 

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10-862395 

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9-995946 

11 

50 

9-134470  1532 

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9-135387  1529 

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1558 

10-860524 

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9-995911 

9 

52 

9-1363031525 

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9-140409 

1555 

10-859591 

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28 

9-995894 

8 

53 

9-1372161522 

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9-141340 

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10-858660 

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9-995876 

7 

54 

9-1381281519 

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9-142269 

1548 

10-857731 

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28 

9-995859 

6 

55 

9-139037  1516 

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30 

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5 

56 

9-139944  1512 

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30 

9-995823 

4 

57 

9-1408501509  -859150 

9-145044 

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28 

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3 

58 

9-1417541506  -858246 

9-145966 

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9-995788 

2 

59 

9-142655  1503 

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9-146885 

153210-853115 

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28 

9-995771 

1 

60 

9-1435551500 

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9-147803 

1529^0-852197 

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30 

9-995753 

0 

'  ||   Cosine.         Secant.   i  Cotangent. 

Tangent.   Cosecant.  1 

Sine. 

' 

82  DEG. 


LOGARITHMIC    SINES,  ETC. 


49 


8  DEG. 


1 

Siue. 

Diff. 

100" 

Cosecant. 

Tangent. 

JJjf-  •  Cotangent. 

Secant 

K*   C,.»ne. 

' 

0 

9-143555 

•856445 

9-147803 

j  10-852  197 

•004247 

9-9U5753 

6~0~ 

1 

9-144453 

1496 

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9-148718 

1526|10-851282 

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30  Vt-9<J57:;A  :>',) 

o 

9-145349 

1493 

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1  523-10-850368 

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3 

9-146243 

1490 

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9-150544 

1520 

10-849456 

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30  9  -995099!  57 

4 

9-147136 

1487 

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9-151454 

1517 

10-848546 

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30  9-9'.)oC,81  '.'I 

5 

9-148026 

1484 

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9-152363 

1514 

10-847637 

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28  9-995664  55 

6 

9-148915 

1481 

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9-153269 

1511 

10-846731 

•004354 

30  9-9U5646  54 

7 

9-149802 

1478 

•850198 

9-154174 

1508 

10-845826 

•004372 

30  9-995628 

53 

8 

9-150686 

1475 

•849314 

9-155077 

1505 

10-844923 

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30  9-995610 

52 

9 

9-151569 

1472 

•848431 

9-155978 

1502 

10-844022 

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32  9-995591 

51 

10 

9-152451 

1469 

•847549 

9-156877 

1499 

10-843123 

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30  9-995573 

50 

11 

9-153330 

1466 

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9-157775 

1496 

10-842225 

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30  9-995555 

49 

12 

9-154208 

1462 

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9-158671 

1493 

10-841329 

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30  :9-995637 

48 

13 

9-155083 

1460 

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9-159565 

1490 

10-840435 

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30  :9-995519 

47 

14 

9-155957 

1457 

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9-160457 

1487 

10-839543 

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30  9-995501 

46 

15 

9-156830 

1454 

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9-161347 

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10-838653 

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9-995482 

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16 

9-157700 

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9-162236 

1481 

10-837764 

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9-995464 

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17 

9-158569 

1448 

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9-163123 

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10-836877 

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9-995446 

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18 

9-159435 

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9-164008 

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10-835992 

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32  9-995427 

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19 

9-160301 

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9-164892 

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10-835108 

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30  9-995409 

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20 

9-161164 

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9-165774 

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32  9-995390 

40 

21 

9-162025 

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9-166654 

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10-833346 

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30  9-995372 

39 

22 

9-162885 

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10-832468 

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23 

9-163743 

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24 

9-164600 

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25 

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9-173070 

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44 

9-181374 

1372 

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9-994935J16 

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9-182196 

1369 

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10-812720 

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32 

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9-183016 

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10-811880 

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9-994896 

14 

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9-183834 

1364 

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10-811042 

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32 

9-994877 

13 

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9-184651 

1361 

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9-189794 

1394 

10-810206 

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33 

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12 

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9-185466 

1359 

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9-190629 

1391 

10-809371 

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32 

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11 

50 

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1356 

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10-808538 

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33 

9-994818 

10 

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1386 

10-807706 

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9-994798 

9 

52 

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1351 

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1384 

10-806876 

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32 

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8 

53 

9-188712 

1348 

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9-193953 

1381 

10-806047 

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33 

9-994759 

7 

54 

9-189519 

1346 

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9-194780 

1379 

10-805220 

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33 

9-994739 

6 

55 

9-190325 

1343 

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9-195606 

1376 

10-804394 

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32 

9-994720 

5 

56 

9-191130 

1341 

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9-196430 

1374 

10-803570 

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33 

9-994700 

4 

57 

9-191933 

1338 

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9-197253 

1371 

10-802747 

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33 

9-994680 

3 

58 

9-192734 

1336 

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9-198074 

1369 

10-801926 

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33 

9-994660 

2 

59 

9-193534 

1333 

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9-198894 

1366 

10-801106 

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33 

9-994640 

1 

60 

9-194332 

1330 

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9-199713 

1364 

10-800287 

•005380 

33 

9-994620 

0 

'     Cosine. 

Secant. 

Cotangent. 

Tangent.   Cosecant. 

Sine.   ''  ' 

81   Diiti. 


50                LOGARITHMIC  SINES,  ETC. 

9  DEG. 

' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff 
100' 

Cosine. 

' 

0 

9-194332 

•805668 

9-199713 

.  110-800287 

•005380 

9-994620 

60 

1 

9-195129 

1328 

•804871 

9-200529 

1361J10-799471 

•005400 

33 

9-994600 

59 

2 

9-195925 

1326 

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9-201345 

1359 

10-798655 

•005420 

33 

9-994580 

58 

3 

9-196719 

1323 

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9-202159 

1356 

10-797841 

•005440 

33 

9-994560 

57 

4 

9-197511 

1321 

•802489 

9-202971 

1354 

10-797029 

•005460 

33 

9-994540 

56 

5 

9-198302 

1318 

•801698 

9-203782 

1352 

10-796218 

•005481 

35 

9-994519 

55 

6 

9-199091 

1316 

•800909 

9-204592 

1349 

10-795408 

•005501 

33 

9-994499 

54 

7 

9-199879 

1313 

•800121 

9-205400 

1347 

10-794600 

•005521 

33 

9-994479 

53 

8 

9-200666 

1311 

•799334 

9-206207 

1345 

10-793793 

•005541 

33 

9-994459 

52 

9 

9-201451 

1308 

•798549 

9-207013 

1342 

10-792987 

•005562 

35 

9-994438 

51 

10 

9-202234 

1306 

•797766 

9-207817 

1340 

10-792183 

•005582 

33 

9-994418 

50 

11 

9-203017 

1304 

•796983 

9-208619 

1338 

10-791381 

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33 

9-994398 

49 

12 

9-203797 

1301 

•796203 

9-209420 

1335 

10-790580 

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35 

9-994377 

48 

13 

9-204577 

1299 

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9-210220 

1333 

10-789780 

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33 

9-994357 

47 

14 

9-205354 

1296 

•794646 

9-211018 

1331 

10-788982 

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35 

9-994336 

46 

15 

9-206131 

1294 

•793869 

9-211815 

1328 

10-788185 

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33 

9-994316 

45 

16 

9-206906 

1292 

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9-212611 

1326 

10-787389 

•005705 

35 

9-994295 

44 

17 

9-207679 

1289 

•792321 

9-213405 

1324 

10-786595 

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35 

9-994274 

43 

18 

9-208452 

1287 

•791548 

9-214198 

1321 

10-785802 

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33 

9-994254 

42 

19 

9-209222 

1285 

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9-214989 

1319 

10-785011 

•005767 

35 

9-994233 

41 

20 

9-209992 

1282 

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9-215780 

1317 

10-784220 

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35 

9-994212 

40 

21 

9-210760 

1280 

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9-216568 

1315 

10-783432 

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35 

9-994191 

39 

22 

9-211526 

1278 

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9-217356 

1312 

10-782644 

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33 

9-994171 

38 

23 

9-212291 

1275 

•787709 

9-218142 

1310 

10-781858 

•005850 

35 

9-994150 

37 

24 

9-213055 

1273 

•786945 

9-218926 

1308 

10-781074 

•005871 

35 

9-994129 

36 

25 

9-213818 

1271 

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9-219710 

1305 

10-780290 

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35 

9-994108 

35 

26 

9-214579 

1268 

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9-220492 

1303 

10-779508 

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35 

9-994087 

34 

27 

9-215338 

1266 

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9-221272 

1301 

10-778728 

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35 

9-994066 

33 

28 

9-216097 

1264 

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9-222052 

1299 

10-777948 

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35 

9-994045 

32 

29 

9-216854 

1261 

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9-222830 

1297 

10-777170 

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35 

9-994024 

31 

30 

9-217609 

1259 

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9-223607 

1294 

10-776393 

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9-994003 

30 

31 

9-218363 

1257 

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9-224382 

1292 

10-775618 

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35 

9-993982 

29 

32 

9-219116 

1255 

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9-225156 

1290 

10-774844 

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37 

9-993960 

28 

33 

9-219868 

1253 

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9-225929 

1288  10-774071 

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35 

9-993939 

27 

34 

9-220618 

1250 

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9-226700 

1286 

10-773300 

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35 

9-993918 

26 

35 

9-221367 

1248 

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9-227471 

1284 

10-772529 

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35 

9-993897 

25 

36 

9-222115 

1246 

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9-228239 

1281  10-771761 

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37 

9-993875 

24 

37 

9-222861 

1244 

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9-229007 

1279 

10-770993 

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35 

9-993854 

23 

38 

9-223606 

1242 

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9-229773 

1277 

10-770227 

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37 

9-993832 

22 

39 

9-224349 

1239 

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9-230539 

1275 

10-769461 

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35 

9-993811 

21 

40 

9-225092 

1237 

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9-231302 

1273 

10-768698 

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37 

9-993789 

20 

41 

9-225833 

1235 

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9-232065 

1271 

10-767935 

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35 

9-993768 

19 

42 

9-226573 

1233 

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9-232826 

1269 

10-767174 

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37 

9-993746 

18 

43 

9-227311 

1231 

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9-233586 

1267 

10-766414 

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35 

9-993725 

17 

44 

9-228048 

1228 

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9-234345 

1265 

10-765655 

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37 

9-993703 

16 

45 

9-228784 

1226 

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9-235103 

1262 

10-764897 

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37 

9-993681 

15 

46 

9-229518 

1224 

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9-235859 

1260 

10-764141 

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35 

9-993660 

14 

47 

9-230252 

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9-236614 

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9-993638 

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9-230984 

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9-231715 

1218 

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9-233899 

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9-238953  1197 

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10-753681 

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9-993351 

0 

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80  DEG. 


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9-242526 

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9-243237 

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9-249998 

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9-243947 

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9-244656 

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9-245363 

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9-246069 

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9-246775 

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46 

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9-250282 

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10-742731 

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9-250980 

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17 

9-251677 

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10-741290 

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9-992967,43 

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10-740571 

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0 

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' 

35 


79  DEC. 


LOGARITHMIC    SINES,  ETC. 


11  DEG. 


/ 

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Diff. 
100" 

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100" 

Cosine.     ' 

0  9-280599 

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9-288652 

10-711348 

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9-991947  160 

1 
2 

9-281248 
9-281897 

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9-991848|  56 

5 

9-283836 

1076 

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9-292013 

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10-707987 

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9-991823  55 

6 

19-284480 

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10-707318 

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9-991799154 

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9-285766 

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9-991749 

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9-286408 

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51 

10 

9-287048 

1067 

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78  DEO. 

* 

LOGARITHMIC    SINES,  ETC. 


12  DEG. 


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100" 

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100" 

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gjf-   Cosine. 

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•597995  9-416293 

863 

10-583707 

•014288 

55 

9-985712 

23 

38 

9-402489  807  -597511  9-416810 

862  10-583190 

•014321 

55 

9-985679 

22 

39 

9-402972!  806  -597028  I  9-417326 

861 

10-582674 

•014354 

55  9-985646 

21 

40! 

9-403455!  805 

•596545  9-417842 

860 

10-582158 

•014387 

55  9-985613 

20 

41 

9-4039381  804 

•596062  9-418358 

859 

10-581642 

•014420 

55  (9-985580 

19 

42 

9-404420J  803 

•595580  9-418873 

858 

10-581127 

•014453 

55 

9-985547 

18 

43 
44 

9-404901  802 
9-405382  801 

•595099  9-419387 
•594618  9-419901 

857 
856 

10-580613 
10-580099 

•014486 
•014520 

55 
57 

9-985514 

9-985480 

17 
16 

45 

9  -405862  1  800 

•594138  !!  9-420415 

855)10-579585 

•014553 

55 

9-985447 

15 

46 

9-4063411  799 

•593659  i  9-420927 

855 

10-579073 

•014586 

55 

9-985414 

14 

47 

9-406820]  798 

•593180  ii  9-421440 

854 

10-578560 

•014619  5519-985381 

13 

48 

9-407299 

797 

•592701  ;;  9-421952 

853 

10-578048 

•014653 

57  {9-985347 

12 

49 

9-407777 

796 

•592223  M  9-422463 

852 

10-577537 

•014686 

55  9-985314 

11 

50 

9-408254 

795 

•591746  19-422974 

851 

10-577026 

•014720!  57  9-985280 

10 

51 

9-408731 

794 

•591269 

9-423484 

850 

10-576516 

•0147531  5519-985247 

9 

52 

9-409207 

794 

•590793 

9-423<f3 

849 

10-576007 

•014787 

57  J9-985213 

8 

53 

9-409682 

793 

•590318 

9-4245S3 

848 

10-575497 

•014820 

55  9-985180 

7 

54 

9-410157 

792 

•589843  |j  9-425011 

848 

10-574989 

•014854  57 

9-985146 

6 

55 

9-410632 

791 

•589368  1  9-425519 

847 

10-574481 

•014887 

55 

9-985113 

5 

56 

9-411106 

790 

•588894!!  9-426027 

846 

10-573973 

014921 

57  .'9-985079 

4 

57 

9-411579 

789 

•588421  !:  9-426534 

845 

10-573466 

014955 

57  9-985045 

3 

58 

9-412052 

788 

•587948  9-427041 

844 

10-572959  -014989 

57  9-985011 

2 

59 

9-412524 

787 

•587476  '9-427547 

843 

10-572453  -015022 

55 

9-984978  1 

60 

9-412996 

786 

•587004  i  9-42805,2 

843 

10-571948  -015056 

57 

9-984944  0 

Cosine. 

Secant.   ,:  Cotangent. 

Tangent.   Cosecant. 

Sine.     ' 

75  DEG. 


56 


LOGARITHMIC    SINES,  ETC. 


15  DEC. 


r 

Sine. 

Diff. 
100" 

Cosecant.    ,|    Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine.  ( 

• 

0 

9-412996 

•587004    9-428052 

10-571948 

•015056 

9-984944  lift 

1 

9-413467 

785 

•586533  ;  9-428557 

842 

10-571443 

•015090 

57 

9-984910)  159 

2 

9-413938 

784 

•586062    9-429062 

841 

10-570938 

•015124 

57 

9-984876158 

3 

9-414408 

783 

•585592    9-429566 

840 

10-570434 

•015158 

57  9-984842 

57 

4 
5 

9-414878 
9-415347 

783 

782 

•585122 
•584653  ! 

9-430070 
9-430573 

839 
838 

10-569930 
10-569427 

•015192 
•015226 

57  9-984808,56 
57  9-984774J55 

"6 

9-415815 

781 

•584185  :  9-431075 

838 

10-568925 

•015260 

57  I9-984740J  54 

7 

9-416283 

780 

•583717  19-431577 

837 

10-568423 

•015294 

57  9  -984706!  |53 

8 

9-416751 

779 

•583249    9-432079 

836 

10-567921 

•015328 

57  9-984672  52 

9 

9-417217 

778 

•582783  19-432580 

865 

10-567420 

•015362 

57  19-984638  51 

10 

9-417684 

777 

•582316  i  9  -433080 

834 

10-566920 

•015397 

58  |9-984603  50 

11 

9-418150 

776 

•581850    9-433580 

833 

10-566420 

•015431 

57  9-984569'  49 

12 

9-418615 

775 

•581385:  9-434080 

832 

10-565920 

•015465 

57  9-984535 

48 

13 

9-419079 

774 

•580921    9-434579 

832 

10-565421 

•015500 

58  9-984500147 

14 
15 

9-419544 
9-420007 

773 
773 

•580456  J  9-435078 
•579993  '  9-435576 

831 

830 

10-564922 
|P  -564424 

•015534 
•015568 

57  9-984466J46 
57  9-984432  45  , 

16 

9-420470 

772 

•579530  i 

9-436073 

829 

10-563927 

•015603 

58 

9-984397 

44 

17 

9-420933 

771 

•579067  j 

9-436570 

828 

10-563430 

•015637 

57 

9-984363 

43 

18 

9-421395 

770 

•578605 

9-437067 

828 

10-562933 

•015672 

58 

9-984328 

42 

19 

9-421857 

769 

•578143  ! 

9-437563 

827 

10-562437 

•015706 

58 

9-984294 

41 

20 

9-422318 

768 

•577682 

9-438059 

826 

10-561941 

•015741 

57  9-984259 

40 

21    9-422778 

767 

•577222    9-438554 

825 

10-561446 

•015776 

68 

9-984224! 

39 

22 

9-423238 

767 

•576762    9-439048  824 

10-560952 

•015810 

57 

9-984190; 

38 

231 

9-423697 

766 

•576303    9-439543  823 

10-5G0457 

•015845 

58 

9-984155 

37 

241 

9-424156 

765 

•575844    9-440036 

823 

10-559964 

•015880 

58 

9-984120! 

36 

251 

9-424615 

764 

•575385    9-440529 

822 

10-559471 

•015915 

58  9-984085i 

35 

26 

9-425073 

763 

•574927  i  9-441022 

821 

10-558978 

•015950 

58  9-984050 

34 

27 

9-425530 

762 

•574470    9-441514 

820 

10-558486 

•015985 

58  9-984015; 

33 

28 

9-425987 

761 

•574013  :i  9-442006 

819 

10-557994 

•016019 

57  19-983981  1 

32 

29    9-426443 

30119-426899 

760 
760 

•573557    9-442497 
•573101    9-442988 

819 
818 

10-557503 
10-557012 

•016054 
•016089 

58  9-988946  31 
58  9-983911  30 

31  19-427354 

759  1   -572646    9-443479 

817 

10-556521 

•016125 

60  I9-983875JI29 

32  i  i  9-427809 

758  i  -572191 

9-443968 

816  10-556032 

•016160 

58 

9  -983840  J28 

33    9-428263  757  I   -571737 

9-444458 

816  10-555542  j-016195 

58 

9-983805 

27 

34    9-428717  756  !  -571283 

9-444947 

815  10-555053  [-016230 

58 

9-983770 

26 

35  |9-429170  755:   -570830    9-445435 

814  10-554565  !j-016265  58  9-983735  25 

36  IJ9-429623  754  j   -570377 

9-445923 

813  10-554077    -016300;  58  9-98370024 

37    9-430075  753     -569925 

9-446411 

812  ;iO-553589i  -016336  60  9-983664 

23 

38!  (9-430527  752;  -569473 

9-446898 

812  10-553102 

•016371   58  9-983629 

22 

39  h  9-430978  752  i  -569022 

9-447384 

811  jlO-552616 

•016406!  58  19-983594  21 

40 

9-431429  751     -568571 

9-447870 

810  10-552130 

•01  6442  i  60  9-983558  20 

41 

9-431879  750  !  -568121 

9-448356 

809  '10-551644 

-0164771  58  9-983523: 

19 

42 

9-432329  749     -567671 

9-448841 

809ilO-551159 

•016513J  60  9-983487 

18 

43 
44 

9-432778  749     -567222 
9-433226  748     -566774 

9-449326 
9-449810 

808  10-550674 
807  !  10-5501  90 

•016548  58  9-983452 
-0165841  60  9-983416 

17 
16 

45  19-433675  747 

•566325 

9-450294 

806  10-549706 

•016619  58  9-983381 

15 

46    9-434122  746 

•565878 

9-450777 

806110-549223 

•016655  60  9-983345 

14 

47 

9-434569  745 

•565431  | 

9-451260 

805  110-548740  I-016691]  60(9-983309 

13 

48    9-435016  744 

•564984  1 

9-451743 

804  110-548257   -016727  60  i9-983273 

12 

49  |  9-435462  744 

•564538  i 

9-452225 

803  10-547775 

•0167621  58  9-983238 

11 

50    9-435908  743 

•564092  i 

9-452706 

802  10-547294 

•016798)  60 

9-983202 

10 

51  •  9-436353  742     -563647  i 

9-453187 

802  10-546813 

•0168341  60 

9-983166 

9 

52 

9-436798  741     -563202 

9-453668 

801  10-546332 

•016870J  60 

9-983130 

8 

53 

9-437242.740     -562758 

9-454148 

800 

10-545852 

•016906  60 

9-983094 

7 

54 

9-437686  740 

•562314 

9-454628 

799 

10-545372 

•016942 

60 

9-983058 

6 

55 

9-438129  739 

•561871 

9-455107 

799 

10-544893 

•016978 

60 

9-983022 

5 

56 

9-438572  738 

•561428 

9-455586 

798 

10-544414 

•017014 

60 

9-982986 

4 

67 

9-439014  737 

•560986 

9-456064 

797 

10-543936 

•017050 

60  9-982950 

3 

58    9-439456  736 

•560544    9-456542 

796 

10-543458 

-017086 

60 

9-982914 

2 

59    9-439897  736 

•560103  i  9-457019 

796 

10-542981 

•017122 

60 

9-982878 

1 

60 

9  -440338  735     -559662    9  -457496 

795t 

10-542504 

•017158 

60 

9-982842 

0 

'     1  1      Cosine.                      Secant,      i   Cotangent. 

Tangent.     1  '  Cosecant. 

Sine. 

r~ 

74  UKQ. 


LOGARITHMIC    SINES,  ETC. 


57 


16  BEG. 


/ 

Sine. 

Diff. 
100" 

Cosecant.  ||  Tangent. 

{>$•   Cotangent. 

Secant. 

?oo»;  C"™*-   ' 

0 

1 

9-440338 

9-440778 

734 

•559662  j 
•559222 

9-457496 
9-457973 

10-542504 
794  [10-542027 

017158 

017195 

9-982842  160 
62  9-982805  59 

2 

9-441218 

733 

•5587821  i9-458449 

793  110-541551 

017231 

60  9-982769  58 

3 

9-441658 

732 

•558342 

9-458925 

793  10-541075 

017267 

60  9-982738  57 

4 

9-442096 

731 

•557904 

9-459400 

792 

10-540600 

017304 

62  9-<J82696:56  ! 

5 

9-442535 

731 

•557465 

9-459875 

791 

10-540125 

017340 

60  9-9826«iO!55  j 

6 

9-442973 

730 

•557027 

9-460349 

790 

10-539651 

017376 

60  9-982624S54 

7 

9-443410 

729 

•556590 

9-460823 

790 

10-539177 

017413 

62  9  -982587:  ;53 

8 

9-443847 

728 

•556153 

9-461297 

789 

10-538703 

017449 

60  9-982551  ;52 

9 

9-444284 

727 

•555716 

9-461770 

788 

10-538230 

017486 

62  9-982514 

51 

10 

9-444720 

727 

•555280 

9-462242 

788 

10-537758 

017523 

62  9-982477 

50 

11 

9-445155 

726 

•554845 

9-462714 

787 

10-537286 

017559 

60  9-982441  49 

12 

9-445590 

725 

•554410 

9-463186 

786 

10-536814 

017596 

62  9-982404)48 

13 

9-446025 

724 

•553975 

9-463658 

785 

10-536342 

017633 

62  9-982367 

47 

14 

9-446459 

723 

•553541 

9-464128 

785 

10-535872 

017669 

60  9-982331 

46 

15 

9-446893 

723 

•553107 

9-464599 

784 

10-535401 

017706 

62  9-982294  |45 

16 

9-447326 

722 

•552674 

9-465069 

783 

10-534931 

017743 

62  9-982257l!44 

17 

9-447759 

721 

•552241 

9-465539 

783 

10-534461 

017780 

62  9-982220  43 

18 

9-448191 

720 

•551809 

9-466008 

782 

10-533992 

017817 

62  9-982183  42 

19 

9-448623 

720 

•551377  ji  9-466476 

781 

10-533524 

017854 

62  9-9821415.41 

20 

9-449054 

719 

•550946 

9-466945 

780 

10-533055 

017891 

62  9-982109  40 

21 

9-449485 

718 

•550515 

9-467413 

780 

10-532587 

017928 

62  9-982072 

39 

22 

9-449915 

717 

•550085 

9-467880 

779 

10-532120 

•017965 

62  9-982035 

38 

23 

9-450345 

716 

•549655 

9-468347 

778 

10-531653 

•018002 

62  9-981998 

37 

24 

9-450775 

716 

•549225 

9-468814 

778 

10-531186 

•018039 

62  9-981961 

36 

25 

9-451204 

715 

•548796 

9-469280 

777 

10-530720 

•018076 

62  9-981924 

35 

26 

9-451632 

714 

•648368 

9-469746 

776 

10-530254 

•018114 

63  9-981886  34 

27 

9-452060 

713 

•547940 

9-470211 

775 

10-529789 

•018151 

62  9-981849 

33 

28 

9-452488 

713 

•547512 

9-470676 

775 

10-529324 

•018188 

62  9-981812 

32  1 

29 

9-452915 

712 

•547085 

•9-471141 

774 

10-528859 

•018226 

63  9-981774 

31 

30 

9-453342 

711 

•546658 

9-471605 

773 

10-528395 

•018263!  62  9-981737 

30 

31 

9-453768 

710 

•54G232 

i  9-472068 

773  110-527932 

•018300 

62  9-981700  29 

32 

9-454194 

710 

•5458061!  9-472532 

772  ilO-527468 

•018338 

63 

9-981662  28 

33 

9-454619 

709 

•545381  9-472995 

771  110-527005 

•018375 

62 

9-981625|  27 

34 

9-455044 

708 

•544956 

9-473457 

771  ilO-526543 

•018413 

63 

9-9815871  26 

35 

9-455469 

707 

•544531 

9-473919 

770110-526081 

•018451 

63 

9-981549  25 

36 

9-455893 

707 

•544107 

9-474381 

769110-525619 

•018488 

62 

9-981512  24 

37 

9-456316 

706 

•543684 

j  9  -474842 

769 

10-525158 

•018526 

63 

9-98l474;23  1 

38 

9-456739 

705 

•543261 

9-475303 

768 

10-524697 

•018564 

63 

9-981436  22  ! 

39 

9-457162 

704 

•542838 

9-475763 

767 

10-524237 

•018601 

62 

9-981399  21  i 

40 

9-457584 

704 

•542416  9-476223 

767 

10-523777 

•018639 

63 

9-981361  20  ! 

41 

9-458006 

703 

•5419941  9-476683 

766 

10-523317 

•018677 

63 

9-981323  ;19 

42 

9-458427 

702 

•541573  9-477142 

765 

10-522858 

•018715 

63 

9-981285  18 

43 

9-458848 

701 

•541152 

9-477601 

765 

10-522399 

•018753 

63 

9-981247  17 

44 

9-459268 

701 

•540732 

9-478059 

764 

10-521941 

•018791 

63 

9-981209  16 

45 

9-459688 

700 

•540312 

9-478517 

763 

10-521483 

•018829 

63 

9-981171 

15 

46 

9-460108 

699 

•539892 

9-478975 

763 

10-521025 

•018867 

63 

9-981133 

14 

47 

9-460527 

698 

•539473 

9-479432  762 

10-520568 

•018905 

63 

9-981095 

13 

48 

9-460946 

698 

•539054  |j  9-479889  761 

10-520111 

•018943 

63 

9-981057 

12 

49 
50 

9-461364 
9-461782 

697 
696 

•538636!  9-480345  761 
•538218  !l  9-480801  760 

10-519655  -018981 
10-519199:  -019019 

63 
63 

9-981019 
9-980981 

11 
10 

51 

9-462199 

695 

•537801 

9-481257 

759 

10-518743  -019058  65 

9-980942 

9 

52 

9-462616 

695 

•537384 

9-481712 

759 

10-518288!  -019096!  63 

9-980904 

8 

53 

9-463032 

694 

•536968 

9-482167 

758 

10-517833  i  -019134  63 

9-980866 

7 

54 

9-463448 

693 

•536552 

9482621  757 

10-517379  -019173  65 

9-980827 

6 

55 

9-463864 

693 

•536136 

9-4830751  757 

10-516925  -019211  63 

9-980789 

5 

56 

9-464279 

692 

•535721 

9-483529,  756 

10-516471  -019250'  65 

9-980750 

4 

57 

9-464694 

691 

•535306 

i  9-483982 

755 

10-516018  1-019288;  63  9-980712 

3 

58 

9-465108 

690 

•534892 

9-484435 

755 

10-515565  -019327  65 

9-9801)73 

2 

59 

9-465522 

690 

•534478 

;  9-484887 

754 

10-515113  -019365  63 

9-980635 

1 

60 

9-465935 

689 

•534065 

9-485339  753 

10-514661  -019404  65 

9-980596 

0 

i 

Sine. 

'  ' 

Cosine. 

Secant 

,  Cotangent. 

Tangent.   Cosecant. 

!  ' 

73  UEG. 


58 


LOGARITHMIC    SINES,  ETC. 


17  D EG. 


' 

Sine. 

?«?'   C°8ecant- 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Dift. 

100" 

Cosine.  |  ' 

0 

9-4(55935 

•534065 

9-485339 

10-514661 

•019404 

9-980596  60 

1 

9-466348 

688 

•533652 

9-485791 

753 

10-514209 

•019442 

63 

9-980558 

59 

2 

9-466761 

688 

•533239 

9-486242 

752 

10-513758 

•019481 

65 

9-98051  9  '58 

3 

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9-978206 

0 

1  ' 

Cosine. 

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Cosecant. 

Sine. 

' 

72  DKG. 


.    LOGARITHMIC    SINES,  ETC. 


59 


18  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

' 

Q~ 

9-489982 

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9-511776 

10488224 

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60 

1 

9-490371 

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9-491535 

646 

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9-978042 

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9-978001 

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6 

9-492308 

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0 

!   Cosine.   I        Secant.   .  Cotangent. 

Tangout. 

Cosecant. 

Sine. 

' 

71 


60 


LOGARITHMIC    SINES,  ETC. 


19  DEG. 


Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff 

100" 

Cosine. 

' 

0 

9-512642 

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10-463028 

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593 

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9-550752 

667 

10-449248 

•025833 

75 

9-974167 

26 

35 

9-525275 

593 

•474725 

9-551152 

667 

10-448848 

•025878 

75 

9-974122 

25 

36 

9-525630 

592 

•474370 

9-551552 

666 

10-448448 

•025923 

75 

9-974077 

24 

37 

9-525984 

591 

•474016 

9-551952 

666 

10-448048 

•025968 

75 

9-974032 

23 

38 

9-526339 

591 

•473661 

9-552351 

665 

10-447649 

•026013 

75 

9-973987 

22 

39 

9-526693 

590 

•473307 

9-552750 

665. 

10-447250 

•026058 

75 

9-973942 

21 

40 

9-527046 

590 

•472954 

9-553149 

665 

10-446851 

•026103 

75 

9-973897 

20 

41 

9-527400 

589 

•472600 

9-553548 

664 

10-446452 

•026148 

75 

9-973852 

19 

42 

9-527753 

589 

•472247 

9-553946 

664 

10-446054 

•026193 

75 

9-973807 

18 

43 

9-528105 

588 

•471895 

9-554344 

663 

10-445656 

•026239 

75 

9-973761 

17 

44 

9-528458 

588 

•471542 

9-554741 

663 

10-445259 

•026284 

75 

9-973716 

16 

45 

9-528810 

587 

•471190 

9-555139 

662 

10-444861 

•026329 

76 

9-973671 

15 

46 

9-529161 

587 

•470839 

9-555536 

662 

10-444464 

•026375 

76 

9-973625 

14 

47 

9-529513 

586 

•470487 

9-555933 

661 

10-444067 

•026420 

76 

9-973580 

13 

48 

9-529864 

586 

•470136 

9-556329 

661 

10-443671 

•026465 

76 

9-973535 

12 

49 

9-530215 

585 

•469785 

9-556725 

660 

10-443275 

•026511 

76 

9-973489 

11 

50 

9-530565 

585 

•469435 

9-557121 

660 

10-442879 

•026556 

76 

9-973444 

10 

51 
52 

9-530915 
9-531265 

584 
584 

•469085 
•468735 

9-557517 
9-557913 

659  10-442483 
659  10-442087 

•026602 
026648 

76 
76 

9-973398 
9-973352 

9 
8 

53 

9-531614 

583 

•468386 

9-558308 

659 

10-441692 

026693 

76 

9-973307 

7 

54 

9-531963 

582 

•468037 

9-558702 

668 

10-441298 

026739 

76 

9-973261 

6 

55 

9-532312 

582 

•467688 

9-559097 

658  10-440903 

026785 

76 

9-973215 

5 

56 

9-532661 

581 

•467339 

9-559491 

657  10-440509 

026831 

76 

9-973169 

4 

57 

9-533009 

581 

•466991 

9-559885 

657  10-440115 

026876 

76 

9-973124 

3 

58 

9-533357 

580 

•466643 

9-560279 

656  ilO-439721 

026922 

76  9-973078 

2 

5'J 

9-533704  580 

•466296 

9-560673 

656  10-439327 

026968 

76  :9-973082 

1 

60 

9-534052  579 

•465948 

9-561066 

655  10-438934 

027014  76  i9-972986| 

0 

'   '   Cosine.         Secant.   '  Cotangent. 

Tangent. 

Cosecant.  ;       Sine.    ' 

70  DEG. 


LOGARITHMIC    SINES,  ETC. 


61 


20  DEC. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 

100" 

Cosine,  j  / 

0 

9-534052 

•465948 

9-561066 

10-438934 

•027014 

9  -972986  |60 

1 

9-534399 

578 

•465601 

9-561459 

655 

10-438541 

•027060 

77 

9-972940  69 

2 

9-534745 

577 

•465255 

9-561851 

654 

10-438149 

•027106 

77 

9-972894  158 

3 

9-535092 

577 

•464908 

9-562244 

654 

10-437756 

•027152 

77 

9-972848 

57 

4 

9-535438 

577 

•464562 

9-562636 

653 

10-437364 

•027198 

77 

9-972802  56 

5 

9-535783 

576 

•464217 

!  9-563028 

653 

10-436972 

•027245 

78  9-972755  55 

6 

9-536129 

576 

•463871 

'  9-563419 

653 

10-436581 

•027291 

77  9-972709  54 

7 

9-536474 

575 

•463526 

9-563811 

652 

10-436189 

•027337 

77  9-972663 

53 

8 

9-536818 

574 

•463182  9-564202 

652 

10-435798 

•027383 

77  9-972617 

52 

9 

9-537163 

574 

•462837 

9-564592 

651 

10-435408 

•027430 

78  9-972570 

51 

10 

9-537507 

573 

•462493 

i  9-564983 

651 

10-435017 

•027476 

77  9-972524  50 

11 

9-537851 

573 

•462149  ||  9-565373 

650 

10-434627 

•027522 

77  9-972478 

49 

12 

9-538194 

572 

•461806  Ij  9-565763 

650 

10-434237 

•027569 

78  :9-972431 

48 

13 

9-538538 

572 

•461462 

9-566153 

649 

10-433847 

•027615 

77  9-972385 

47 

14 

9-538880 

571 

•461120 

9-566542 

649 

10-433458 

•027662 

78  9-972338 

46 

15 

9-539223 

571 

•460777 

9-566932 

649 

10-433068 

•027709 

78  9-972291 

45 

16 

9-539565 

570 

•460435 

9-567320 

648 

10-432680 

•027755 

77  9-972245 

44 

17 

9-539907 

570 

•460093 

9-567709 

648 

10-432291 

•027802 

78  9-972198 

43 

18 

9-540249 

569 

•459751 

9-568098 

647 

10-431902 

•027849 

78  9-972151 

42 

19 

9-540590 

569 

•459410 

9-568486 

647 

10-431514 

•027895 

77  9-972105 

41 

20 

9-540931 

568 

•459069 

9-568873 

646 

10-431127 

•027942 

78  9-972058 

40 

21 

9-541272 

568 

•458728 

9-569261 

646 

10-430739 

•027989 

78  9-972011 

39 

22 

9-541613 

567 

•458387 

9-569648 

645 

10-430352 

•028036 

78  9-971964 

38 

23 

9-541953 

567 

•458047 

9-570035 

645 

10-429965 

•028083 

78  9-971917 

37 

24 

9-542293 

566 

•457707 

9-570422 

645 

10-429578 

•028130 

78  9-971870 

36 

25 

9-542632 

566 

•457368 

9-570809 

644 

10-429191 

•028177 

78  9-971823 

35 

26 

9-542971 

565 

•457029 

9-571195 

644 

10-428805 

•028224 

78  9-971776 

34 

27 

9-543310 

565 

•456690 

9-571581 

643 

10-428419 

•028271 

78  9-971729 

33 

28 

9-543649 

564  -456351 

9-571967 

643 

10-428033 

•028318 

78  9-971682 

32 

29 

9-543987 

564  456013 

9-572352 

642 

10-427648 

•028365 

78  9-971635 

31 

30 

9-544325 

563 

•455675 

9-572738 

642 

10-427262 

•028412 

78 

9-971588 

30 

31 

9-544663 

563 

•455337 

9-573123 

642 

10-426877 

•028460 

80 

9-971540 

29 

32 

9-545000 

562 

•455000 

9-573507 

641 

10-426493 

•028507 

78 

9-971493 

28 

33 

9-545338 

562 

•454662 

9-573892 

641 

10-426108 

•028554 

78 

9-971446 

27 

34 

9-545674 

561 

•454326 

9-574276 

640 

10-425724 

•028602 

80 

9-971398 

26 

35 

9-546011 

561 

•453989 

9-574660 

640 

10-425340 

•028649 

78 

9-971351 

25 

36 

9-546347 

560 

•453653 

9-575044 

639 

10-424956 

•028697 

80 

9-971303 

24 

37 

9-546683 

560 

•453317 

9-575427 

639 

10-424573 

•028744 

78 

9-971256 

23 

38 

9-547019 

559 

•452981 

9-575810 

639 

10-424190 

•028792 

80 

9-971208 

22 

39 

9-547354 

559 

•452646 

9-576193 

638 

10-423807 

•028839 

78 

9-971161 

21 

40 

9-547689 

558 

•452311 

9-576576 

638 

10-423424 

•028887 

80 

9-971113 

20 

41 

9-548024 

558 

•451976 

9-576958 

637 

10-423042 

•028934 

78 

9-971066  19 

42 

9-548359 

557 

•451641 

9-577341 

637 

10-422659 

•028982 

80 

9-971018  18 

43 

9-548693 

557 

•451307 

9-577723 

636 

10-422277 

•029030 

80 

9-970970  17 

44 

9-549027 

556 

•450973 

9-578104 

636 

10-421896 

•029078 

80 

9-970922:16 

45 

9-549360 

556 

•450640 

9-578486 

636 

10-421514 

•029126 

80 

9-970874 

15 

46 

9-549693 

555 

•450307 

'9-578867 

635 

10-421133 

•029173 

78 

9-970827 

14 

47 

9-550026 

555 

•449974 

9-579248 

635 

10-420752 

•029221 

80 

9-970779 

13 

48 

9-550359 

554 

•449641 

9-579629 

634 

10-420371 

•029269 

80 

9-970731 

12 

49 

9-550692 

554 

•449308 

9-580009 

634 

10-419991 

•029317 

80 

9-970683 

11 

50 

9-551024 

553 

•448976 

9-580389 

634 

10-419611 

•029365 

80 

9-970635 

10 

51 

9-551356 

553 

•448644 

9-580769 

633 

10-419231 

•029414 

82 

9-970586 

9 

52 

9-551687 

552 

•448313 

9-581149 

633 

10-418851 

•029462 

80 

9-970538 

8 

53 

9-552018 

552 

•447982 

9-581528 

632 

10-418472 

•029510 

80 

9-970490 

7 

54 

9-552349 

552 

•447651 

9-581907 

632 

10-418093 

•029558 

80 

9-970442 

6 

55 

9-552680 

551 

•447320 

9-582286 

632 

10-417714 

•029606 

80 

9-970394 

5 

56 

9-553010 

551 

•446990 

9-582665 

631 

10-417335 

•029655 

82 

9-970345 

4 

57 

9-553341 

550 

•446659 

9-583043 

631 

10-416957 

•029703 

80 

9-970297 

3 

58 

9-553670 

550 

•446330 

9-583422 

630 

10-416578  -029751 

80 

9-970249 

2 

59 

9-554000 

549 

•446000 

9-583800 

630 

10-416200  -029800 

82 

9-970200 

1 

60 

" 

9-554329 

549 

•445671 

9-584177 

629 

10-415823  -029848 

80 

9-970152 

0 

Cosine. 

Secant.    Cotangent. 

Tangent.   Cosecant. 

Sine. 

' 

DE«. 


62 


LOGARITHMIC    SINES,  ETC. 


21   I)  EG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent.   j^J   Cotangent. 

Secant. 

ig?   Cosine.   j  • 

0 

9-554329 

445671 

9-584177J    10415823 

^029848 

!  9  -9701  52  60 

1 

9:5546581  548 

445342 

9-584555;  629  10415445  -029897  81  9-970103  59 

2 

9-554987!  548 

445013 

9-584932'  629  10415068  -029945 

81  19-970055  58 

3 

9-555315  547 

444685 

9-585309!  628  10414691  il-029994 

81  9-970006  57 

4 

9-555643  547 

•444357 

9-585686  628  10414314  !|-030043 

81  9-969957:  !56 

5 

9-555971  546 

444029 

9-586062!  627  '10413938  -030091 

81  J9-969909:  ;55 

6 

9-556299  546 

443701 

9-5864391  627  10413561  -030140 

81  19-969860;  :54 

7 

9-556626  545 

•443374 

9-586815!  627  10413185  -030189!  81  19-969811!  53 

8i|9-556953i  545 

443047 

9-587190  026  10412810  1-030238!  81  (9-969762  52 

9  ;  9-557280;  544 

442720 

9-587566  626  10412434  !  -0302861  81  9-969714  51 

10  i  9-557606.  544 

•442394 

9-587941  ;  625  10412059 

•030335!  81 

9-969665  50 

11  9-557932:  543 

442068 

9-588316  625  10411684 

•030384 

81 

9-969616  49 

12  1  9-558258!  543 

441742 

9-588691  625  10411309  '-030433 

82  9-969567  ;  J48 

13 

9-558583  543 

441417  9-589066  624  10410934  -030482 

82 

9-969518H47 

14 

9-558909  542 

441091 

9-689440  624  10410560  -030531 

82 

9-969469  J46 

15 

9-559234  542 

•440766 

9-589814'  623  :10410186  '  -030580 

82  !9-969420  45 

16 

9-5595581  541 

•440442 

9-5901881  623  '10409812  i  -030630!  82  19-96937044 

17 

9-559883  541 

•440117 

9-590562  623  10409438 

•0306791  82 

9-969821  43 

18 

9-560207  540 

•439793 

9-590935  622  10409065 

•0307281  82 

9-969272:i42 

19 

9-560531  540 

439469 

9-591308  622  10408692  1-030777  82  9-969223  41 

20 

9-560855  539 

-439145 

9-591681  £22!10408319  -030827  82  19-96917340 

21 

9-561178  539 

•438822  9-592054  621 

10407946  -030876  82  9-969124  39 

22 

9-561501  538  -438499  9-592426 

621 

10407574  :  -0309251  82  19-969075  88 

23 

9-561824J  538 

•438176 

9-592798 

620  J10407202  !  -030975)  82  19-969025  37 

24 

9-562146  537 

-437854 

9-593171  620 

10406829  1  -031024 

82  19-968976  36 

25 

9-562468 

537 

437532 

9-593542 

620 

10406458  -031074 

82  19-968926'  35 

26 

9-562790 

536 

437210 

9-593914 

619 

10406086  j  -031  123  !  83 

9-968877  34 

27 

9-563112 

536 

436888 

9-594285 

619  10405715  1-031173!  83 

9  -968827!  33 

28 

9-563433 

536 

436567 

9-594656 

618 

10  4053441!  -03  1223 

83 

9-968777  32 

29 

9-563755 

535 

-436245 

9-595027 

618 

10404973  !i-031272 

83 

9-968728  31 

30 

9-564075 

535 

435925 

9-595398 

618 

10404602  -031322 

83 

9-968678;  30 

31 

9-564396 

534 

435604 

9-595768 

617 

10404232  I-031372 

83 

9-968628!  29 

32 

9-564716 

534 

435284 

9-596138 

617 

10403862 

•031422 

83 

9-968578128 

33 

9-565036 

533 

•434964 

9-596508 

616 

10-403492 

•031472 

83 

9-968528;  27 

34 

9-565356 

533 

434644 

9-596878 

616 

10403122 

•031521 

83 

9-968479 

26 

35 

9-565676 

532 

434324 

9-597247 

616 

10402753 

•031571 

83 

9-968429 

25 

36 

9-565995 

532 

434005 

9-597616 

615 

10402384 

•031621 

83 

9-968379 

24 

37 

9-566314 

531 

433686 

9-597985 

615 

10402015 

•031671 

83 

9-968329 

23 

38 

9-566632 

531 

•433368 

9-598354 

615 

10-401646 

•031722 

83 

9-968278 

22 

39 

9-566951 

531 

433049 

9-598722 

614 

10401278 

•031772 

83 

9-968228 

21 

40 

9-567269 

530 

432731 

9-599091 

614 

10-400909 

•031822 

84 

9-968178 

20 

41 

9-567587 

530 

432413 

9-599459 

613 

10-400541 

•031872 

84 

9-968128 

19 

42 

9-567904 

529 

•432096 

9-599827 

613 

10400173 

•031922 

84 

9-968078 

18 

43 

9-568222 

529 

431778 

9-600194 

613 

10-399806 

•031973 

84 

9-968027 

17 

44 

9-568539 

528 

431461 

9-600562 

612 

10-399438 

•032023 

84 

9-967977 

16 

45 

9-568856 

528 

431144 

9-600929 

612 

10-399.071 

•032073 

84 

9-967927 

15 

46 

9-569172 

528 

430828 

9-601296 

611 

10-398704 

•032124 

84 

9-967876 

14 

47 

9-569488 

527 

430512 

9-601662 

611 

10-398338 

•032174 

84 

9-967826 

13 

48 

9-569804 

527 

430196 

9-602029 

611 

10-397971 

•032225 

84 

9-967775 

12 

49 

9-570120 

526 

429880 

9-602395 

610 

10-397605 

•032275 

84 

9-967725 

11 

50 

9-570435 

526 

429565 

9-602761 

610 

10-397239 

•032326 

84 

9-967674 

10 

51 

9-570751 

525 

•429249 

9-603127 

610 

10-396873 

•032376 

84 

9-967624 

9 

52 

9-571066 

525 

428934 

9-603493 

609 

10-396507 

•032427 

84 

9-967573 

8 

53 

9-571380 

524 

428620 

9-603858 

609 

10-396142 

•032478 

84 

9-967522 

7 

54 

9-571695 

524 

428305 

9-6042231  609 

10-395777 

•032529 

85 

9-967471 

6 

55 

9-572009 

523!  427991 

9-604588 

608 

10-395412 

•032579 

85 

9-967421 

5  I 

56 

9-572323 

523  427677 

9-604953 

608 

10-395047 

•032630 

85 

9-967370 

4 

57 

9-572636 

523  427364 

9-605317 

607 

10-394683 

•032681  85 

9-967319 

3 

58 

9-572950 

522  427050 

9-605682 

607 

10-394318 

•032732 

85 

9-967268 

2 

59 

9-573263 

522 

426737 

9-606046 

607 

10-393954 

•032788 

85 

9-967217 

1 

60 

9-573575 

521 

426425 

9-606410 

606 

10-393590 

•032834 

85 

9-967166 

0 

Cosine. 

|   Secant.   H  Cotangent. 

Tangent.   '  Cosecant. 

Sine. 

' 

68  DUG. 


LOGARITHMIC    SINES,  ETC, 


63 


22  BEG. 


' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

'  Cotangent. 

Secant. 

Diff. 

100" 

Cosine. 

' 

~~b~ 

9-573575 

•426425 

9-606410 

10-393590 

•032834 

9-967166 

60 

i 

9-573888 

521 

•426112 

9-606773 

606 

10-393227 

•032885 

85 

9-9671T5 

59 

2 

9-574200 

520 

•425800 

9-607137 

606 

10-392863 

•032936 

85 

9-967064 

58 

3 

9-574512 

520 

•425488 

9-607500 

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9-967013 

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9-574824 

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10-392137 

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9-966961 

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9-575136 

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9-608225 

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10-391775 

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9-966910 

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9-575447 

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9-608588 

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10-391412 

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9-966859 

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7 

9-575758 

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9-608950 

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10-391050 

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9-966808J53 

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9-576069 

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•423931 

9-609312 

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10-390688 

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9-966756 

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9 

9-576379 

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9-609674 

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10-390326 

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9-966705 

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9-576689 

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9-610036 

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10-389964 

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9-966653  50 

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9-576999 

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9-610397 

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10-389603 

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9-966602 

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9-577309 

516 

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9-610759 

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10-389211 

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9-966550 

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13 

9-577618 

516 

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9-611120 

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10-388880 

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9-966499 

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14 

9-577927 

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9-611480 

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10-388520 

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9-966447 

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9-578236 

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9-611841 

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10-388159 

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9-966395 

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9-578545 

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10-387799 

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9-966344 

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9-578853 

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10-387439 

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9-966292 

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9-579162 

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10-387079 

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9-966240 

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10-386719 

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9-966188 

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9-579777 

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9-613641 

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10-386359 

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9-966136 

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9-580085 

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9-966085 

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i  22 

9-580392 

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10-385641 

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9-580699 

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9-581312 

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9-581924 

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9-965772 

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10-383491 

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9-965720 

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9-582535 

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10-383133 

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9-965668 

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9-582840 

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10-380992 

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10-380636 

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9-965301 

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9-965248 

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9-585272 

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10-379924 

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9-965195 

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9-585574 

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10-379568 

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9-965143 

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9-585877 

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9-965090 

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9-586179 

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9-586482 

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10-378503 

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9-964984 

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9-586783 

503 

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9-621852 

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10-378148 

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9-964931 

17 

44 

9-587085 

503 

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9-622207 

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10-377793 

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9-964879 

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9-587386 

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10-377439 

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9-964826 

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9-587688 

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9-622915 

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10-377085 

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9-964773 

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9-587989 

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10-376731 

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9-964720 

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10-376377 

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9-588590 

501 

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10-376024 

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9-964613  11 

50 

9-588890 

500 

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9-624330 

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10-375670 

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9-964560 

10 

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9-589190 

500 

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10-375317 

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9-964507 

9 

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499 

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9-625036 

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10-374964 

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9-964454 

8 

53 

9-589789 

499 

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9-625388 

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10-374612 

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9-964400 

7 

54 

9-590088 

499 

•409912 

9-625741 

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10-374259 

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89 

9-964347 

6 

55 

9-590387 

498 

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9-626093 

587 

10-373907 

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89 

9-964294 

5 

56 

9-590686 

498 

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9-626445 

587 

10-373555 

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89 

9-964240 

4 

57 

9-590984 

497 

•409016 

9-626797 

586 

10-373203 

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89 

9-964187 

3 

58 

9-591282 

497 

•408718 

9-627149 

586 

10-372851 

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89 

9-964133 

2 

59 

9-591580 

497 

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9-627501 

586 

10-372499 

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89 

9-964080 

1 

60 

9-591878 

496 

•408122 

9-627852 

585 

10-372148 

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89 

9-964026 

0 

' 

Cosine. 

I   Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

i 

67  DEC. 


64 


LOGARITHMIC    SINES,  ETC, 


23  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

/ 

0 

9-591878 

•408122 

9-627852 

10-372148 

•035974 

9-964026 

60 

1 

9-592176 

496 

•407824 

9-628203 

585 

10-371797 

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9-963972 

59 

2 

9-592473 

495 

•407527 

9-628554 

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10-371446 

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9-963919 

58 

3 

9-592770 

495 

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9-628905 

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10-371095 

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9-963865 

57 

4 

9-593067 

495 

•406933 

9-629255 

584 

10-370745 

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90 

9-963811 

56 

5 

9-593363 

494 

•406637 

9-629606 

584 

10-370394 

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90 

9-963757 

55 

6 

9-593659 

494 

•406341 

9-629956 

583 

10-370044 

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90 

9-963704 

54 

7 

9-593955 

493 

•406045 

9-630306 

583 

10-369694 

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90 

9-963650 

53 

8 

9-594251 

493 

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9-630656 

583 

10-369344 

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90 

9-963596  52 

9 

9-594547 

493 

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9-631005 

583 

10-368995 

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90 

9-963542!  51 

10 

9-594842 

492 

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9-631355 

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90 

9-963488 

50 

11 

9-595137 

492 

•404863 

9-631704 

582 

10-368296 

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9-963434 

49 

12 

9-595432 

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10-367947 

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9-963379 

48 

13 

9-595727 

491 

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9-632401 

581 

10-367599 

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9-963325 

47 

14 

9-596021 

491 

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9-632750 

581 

10-367250 

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90 

9-963271 

46 

15 

9-596315 

490 

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9-633098 

581 

10-366902 

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90 

9-963217 

45 

16 

9-596609 

490 

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9-633447 

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10-366553 

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9-963163 

44 

17 

9-596903 

489 

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9-633795 

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10«66205 

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9-963108 

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9-597196 

489 

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9-634143 

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10-365857 

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9-963054 

42 

19 

9-597490 

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9-634490 

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10-365510 

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9-962999 

41 

20 

9-597783 

488 

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9-634838 

579 

10-365162 

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91 

9-962945 

40 

21 

9-598075 

488 

•401925 

9-635185 

579 

10-364815 

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91 

9-962890;:39 

22 

9-598368 

487 

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9-635532 

578 

10-364468 

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91 

9-962836!  38 

23 

9-598660 

487 

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9-635879 

578 

10-364121 

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91 

9-962781 

37 

24 

9-598952 

487 

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9-636226 

578 

10-363774 

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91 

9-962727 

36 

25 

9-599244 

486 

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9-636572 

577 

10-363428 

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91 

9-962672 

35 

26 

9-599536 

486 

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9-636919 

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10-363081 

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91 

9-962617 

34 

27 

9-599827 

485 

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9-637265 

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10-362735 

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91 

9-962562 

33 

28 

9-600118 

485 

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9-637611 

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10-362389 

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9-962508 

32 

29 

9-600409 

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10-362044- 

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9-962453 

31 

30 

9-600700 

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9-638302 

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10-361698 

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9-962398 

30 

31 

9-600990 

484 

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9-638647 

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10-361353 

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9-962343 

29 

32 

9-601280 

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9-638992 

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10-361008 

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9-962288 

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9-601570 

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9-639337 

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10-360663 

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9-962233 

27 

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9-601860 

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9-639682 

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10-360318 

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9-962178 

26 

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9-602150 

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9-640027 

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10-359973 

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92 

9-962123 

25 

36 

9-602439 

482 

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9-640371 

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10-359629 

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9-962067 

24 

37 

9-602728 

482 

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9-640716 

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10-359284 

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92 

9-962012 

23 

38 

9-603017 

481 

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9-641060 

573 

10-358940 

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92 

9-961957 

22 

39 

9-603305 

481 

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9-641404 

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10-358596 

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9-961902 

21 

40 

9-603594 

481 

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9-641747 

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10-358253 

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9-961846 

20 

41 

9-603882 

480 

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9-642091 

572 

10-357909 

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92 

9-961791 

19 

42 

9-604170 

480 

.395830 

9-642434 

572 

10-357566 

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92 

9-961735 

18 

43 

9-604457 

479 

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9-642777 

572 

10-357223 

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92 

9-961680 

17 

44 

9-604745 

479 

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9-643120 

572 

10-356880 

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92 

9-961624 

16 

45 

9-605032 

479 

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9-643463 

571 

10-356537 

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93 

9-961569 

15 

46 

9-605319 

478 

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9-643806 

571 

10-356194 

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93 

9-961513 

14 

47 

9-605606 

478 

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9-644148 

571 

10-355852 

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93 

9-961458 

13 

48 

9-605892 

478 

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9-644490 

570 

10-355510 

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93 

9-961402  12 

49 

9-606179 

477 

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9-644832 

570 

10-355168 

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93 

9-961346;ll 

50 

9-606465 

477 

•393535 

9-645174 

570 

10-354826 

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93 

9-961290 

10 

51 

9-606751 

476 

•393249 

9-645516 

570 

10-354484 

•038765 

93 

9-961235 

9 

52 

9-607036 

476 

•392964 

9-645857 

569 

10-354143 

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93 

9-961179 

8 

53 

9-607322 

476 

•392678 

9-646199 

569 

10-353801 

•038877 

93 

9-961123 

7 

54 

9-607607 

475 

•392393 

9-646540 

569 

10-353460 

•038933 

93 

9-961067 

6 

55 

9-607892 

475 

•392108 

9-646881 

568 

10-353119 

•038989 

93 

9-961011 

5 

56 

9-608177 

474 

•391823 

9-647222 

568 

10-352778 

•039045 

93 

9-960955 

4 

57 

9-608461 

474 

•391539 

9-647562 

568 

10-352438 

•039101 

93 

9-960899 

3 

58 

9-608745 

474 

•391255 

9-647903 

567 

10-352097 

•039157 

93 

9-960843 

2 

59 

9-609029 

473 

•390971 

9-648243 

567 

10-351757 

•039214 

94 

9-960786 

1 

60 

9-609313 

473 

•390687 

9-648583 

567 

10-351417 

•039270 

94 

9-960730 

0 

' 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

66  DEG. 


LOGARITHMIC   SINES,  ETC. 


65 


24  DEG. 


1 

Sine. 

Diff. 
100" 

Cosecant.   Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff 
100" 

Cosine. 

' 

~~b~ 

9-609313 

•390687 

9-648583 

10-351417 

•039270 

9-960730 

60~ 

1 

9-609597 

473 

•390403 

9-648923 

566 

10-351077 

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94 

9-960674 

59 

2 

9-609880 

472 

•390120 

9-649263 

566 

10-350737 

•039382 

94 

9-960618 

58 

3 

9-610164 

472 

•389836 

9-649602 

566 

10-350398 

•039439 

94 

9-960561 

57 

4 

9-610447 

472 

••389553 

9-649942 

566 

10-350058 

•039495 

94 

9-960505 

56 

5 

9-610729 

471 

•389271 

9-650281 

565 

10-349719 

•039552 

94 

9-960448 

55 

6 

9-611012 

471 

•388988 

9-650620 

565 

10-349380 

•039608 

94 

9-960392 

54 

7 

9-611294 

470 

•388706 

9-650959 

565 

10-349041 

•039665 

94 

9-960335 

53 

8 

9-611576 

470 

•388424 

9-651297 

564 

10-348703 

•039721 

94 

9-960279 

52 

9 

9-611858 

470 

•388142 

9-651636 

564 

10-348364 

•039778 

94 

9-960222 

51 

10 

9-612140 

469 

•387860 

9-651974 

564 

10-348026 

•039835 

94 

9-960165 

50 

11 

9-612421 

469 

•387579 

9-652312 

563 

10-347688 

•039891 

94 

9-960109 

49 

12 

9-612702 

469 

•387298 

9-652650 

563 

10-347350 

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95 

9-960052 

48 

13 

9-612983 

468 

•387017 

9-652988 

563 

10-347012 

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95 

9-959995 

47 

14 

9-613264 

468 

•386736 

9-653326 

563 

10-346674 

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95 

9-959938 

46 

15 

9-613545 

467 

•386455  |9-653663 

562 

10-346337 

•040118 

95 

9-959882 

45 

16 

9-613825 

467 

•386175 

9-654000 

562 

10-346000 

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95 

9-959825 

44 

17 

9-614105 

467 

•385895 

9-654337 

562 

10-345663 

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95 

9-959768 

43 

18 

9-614385 

466 

•385615 

9-654674 

561 

10-345326 

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95 

9-959711 

42 

19 

9-614665 

466 

•385335 

9-655011 

561 

10-344989 

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95 

9-959654 

41 

20 

9-614944 

466 

•385056  9-655348 

561 

10-344652 

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95 

9-959596 

40 

21 

9-615223 

465 

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561 

10-344316 

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95 

9-959539 

39 

22 

9-615502 

465 

•384498 

9-656020 

560 

10-343980 

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95 

9-959482 

38 

23 

9-615781 

465 

•384219 

9-656356 

560 

10-343644 

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95 

9-959425 

37 

24 

9-616060 

464 

•383940 

9-656692 

560 

10-343308 

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95 

9-959368 

36 

25 

9-616338 

464 

•383662 

9-657028 

559 

10-342972 

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95 

9-959310 

35- 

26  9-616616 

464 

•383384 

9-657364 

559 

10-342636 

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96 

9-959253 

34 

27 

9-616894 

463 

•383106 

9-657699 

559 

10-342301 

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96 

9-959195 

33 

28 

9-617172 

463 

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9-658034 

559 

10-341966 

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96 

9-959138 

32 

29 

9-617450 

462 

•382550 

9-658369 

558 

10-341631 

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96 

9-959080 

31 

30 

9-617727 

462 

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9-658704 

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10-341296 

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96 

9-959023 

30 

31 

9-618004 

462 

•381996 

9-659039 

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10-340961 

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9-958965 

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9-958734 

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9-619386!  460 

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10-339290 

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9-958677 

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9-958619 

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9-619938i  459 

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10-338623 

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9-958561 

22 

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9-620213  459 

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9-958503 

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9-958329 

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10-336625 

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9-958213 

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i  9-621861  457 

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10-336293 

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9-958154 

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550 

10-331987 

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9-957393 

2 

59  !  9-625677  452 

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9-668343 

550 

10-331657 

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9-957335 

1 

60  i  9  -625948  452 

•374052 

9-668672 

550 

10-331328 

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98 

9-957276  |  0 

'  [j   Cosine. 

Secant.,  i  Cotangent. 

Tangent. 

Cosecant. 

Sine.   II  ' 

65  DEG. 


66 


LOGARITHMIC    SINES,  ETC. 


25  DEG. 


' 

Sine. 

^ff;   Cosecant,  j 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine.  |  / 

~~0~ 

9-625948 

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9-668673 

10-331327 

•042724 

9-957276 

1 

9-626219 

451 

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9-669002 

550  10-330998 

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9-957217 

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2 

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4 

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9-956921  J54  ! 

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9-671306 

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9-671634 

547  10-328366 

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9-956744:51 

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9-671963 

547  10-328037 

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9-956684:50 

11 

9-628916 

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9-672291 

547  10-327709 

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9-956625  49 

12 

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9-672619 

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9-956566 

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9-673602 

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10-314388 

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8 

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9-685934  536 

10-314066 

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9-954090 

7 

54 
55 

9-640284 
9-640544 

434 
433 

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9-686255  536 
9-6865771  536 

10-313745 
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102 
102 

9-954029 
9-953968 

6 
5 

56 

9-640804 

433 

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9-963906 

4 

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9-953845 

3 

58 

9-641324 

432 

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9-953783 

2 

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9-641583 

432 

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9-953722 

1 

60 

9-641842 

432  -358158 

9-688182;  534  10-311818  -046340103 

9-953660 

0 

' 

Cosine. 

Secaut. 

Cotangent,  i        Tangent^   Cosecant. 

Sine. 

' 

64  DEO. 


LOGARITHMIC    SINES,  ETC. 


67 


26  PEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Taugeut. 

Diff. 

10U" 

Cotangent. 

Secant. 

5JJ>   Cosine. 

' 

0 

9-641842 

•358158  9-688182 

10-311818 

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9-953660J 

60 

1 

9-642101 

431 

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534 

10-311498 

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103  9-953599J 

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2 

9-642360 

431 

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103  9-953537 

58 

3 

9-642618 

431 

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534 

10-310857 

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103  9-953475 

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4 

9-642877 

430 

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9-689463 

533 

10-310537 

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103  9-953413 

56 

5 

9-643135 

430 

•356865 

9-689783 

533 

10-310217 

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103  9-953352!  55 

6 

9-643393 

430 

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9-690103 

533 

10-309897 

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103  9-953290 

54 

7 

9-643650 

430 

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9-690423 

533 

10-309577 

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103  9-953228 

53 

8 

9-643908 

429 

•356092 

9-690742 

533 

10-309258 

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1039-953166 

52  i 

9 

9-644165 

429 

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9-691062 

532 

10-308938 

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1039-953104 

51 

10 

9-644423 

429 

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9-691381 

532 

10-308619 

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103  9-953042  50 

11 

9-644680 

428 

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9-691700 

532 

10-308300 

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49 

12 

9-644936 

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9-693930 

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10-306070 

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19 

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9-694248 

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10-305752 

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9-646984 

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9-647749 

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10-304482 

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9-648004 

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9-695836 

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10-304164 

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9-648258 

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10-303847 

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521 

10-292834  -0501]  9 

107 

9-949881 

0 

i 

Cosine. 

Secant.    Cotangent. 

Tangent.  II  Cosecant. 

Sine. 

' 

36 


63  DEG. 


C8                 LOGARITHMIC  SINES,  ETC. 

27  PEG. 

' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent.   gjj; 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

i    ! 
/ 

0 

9-657047 

•342953 

9-7071661 

10-292834 

•050119 

9-949881 

eV 

1 

9-657295 

413 

•342705 

9-707478  520 

10-292-.22  j  -050184 

107 

9-949816 

59 

2 
3 
4 
5 
6 

9-657542 

9-057790 
9-658037 
9-658284 
9-658531 

413  -342458 
412[  -312210 
412  -3419ii3 
412  -341716 
412  -341469 

9-707790^  520 
9-708102  520 
9-7084141  520 
9-708726  519 
9-709037  519 

10-292210 
10-291898 
10-291586 
10-291274 
10-290963 

i  -050248 
•050312 
•050377 
1-050442 
•050606 

107 
107 
108 
108 
108 

9-949752'!58 
9-949688  157 
9-949623(156 
9-949558155 
9-949494!  i54 

7 

9-658778 

411  -341222 

9-709349  519  10-290651  i  -050571 

108  9-949429  53 

8 

9-659025 

411 

•340975 

9-709660;  519  110-290340  '-050636 

1089-94986462 

9 

9-659271 

411 

•340729 

9-709971!  519  10-290029  ij-050700ll08  9-949300l!51 

10 

9-659517 

410 

•340483 

9-710282;  518 

10-289718  -050765  108  9-949235^50 

11 

9-659763 

410 

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9-710598  518 

10-289407 

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12 

9-660U09 

410 

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10-289096 

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13 

9-660255 

409 

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108  9-949040  47 

14 

9-6G05U1 

409 

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518 

10-288475 

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15 

9-660746 

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10-288164  1  -051090 

108 

9-948910145 

16 

9-660991 

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10-287854 

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9-948845144 

17 

9-661236 

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10-287544 

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9  -948780!  43 

18 

9-661481 

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10-287234 

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9-948715!42 

19 

9-661726 

408 

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10-286924 

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9-948650J41 

20 

9-661970 

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-338030 

9-713386 

516 

10-286614 

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109 

9-948584  40 

21 

9-662214 

407 

•337786 

9-713696 

516 

10-286304 

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109 

9-948519  39 

22 

9-662459 

407 

•337541 

9-714005 

516 

10-285995 

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109 

9-948454!i38 

23 

9-662703 

407 

•337297 

9-714314 

516 

10-285686 

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109 

9-948388137 

24 

9-662946 

406 

•337054 

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10-285376 

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109 

9-948323 

<36 

25 

9-663190 

406 

•336810 

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10-285067 

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109 

9-948257 

J35 

26 

9-663433 

406 

•336567 

9-715242 

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10-284758 

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109 

9-948192 

|84 

27 

9-663677 

405 

•336323 

9-7155511  515 

10-284449 

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109 

9-948126  33 

28 

9-663920 

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10-284140 

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9-948060  |32 

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10-283832 

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9-947995!31 

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10-283523 

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9-947929 

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9-664648 

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10-283215 

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9-947863 

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32 

9-664891 

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10-282907 

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9-947797 

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33 

9-665133 

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9-665375 

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10-282291 

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9-947665 

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35 

9-665617 

403 

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513 

10-281983 

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9-947600 

25 

36 

9-665859 

403 

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513 

10-281675 

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9-947533 

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37 

9-666100 

402 

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10-281367 

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9-947467 

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38 

9-666342 

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9-947401 

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39 

9-666583 

402 

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9-719248  512 

10-280752 

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110 

9-947335 

21 

40 

9-666824 

402 

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9-719555  512 

10-280445 

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110 

9-947269 

20 

41 

9-667065 

401 

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9-719862 

512 

10-280138 

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110 

9-947203 

19 

42 

9-667305 

401 

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9-720169 

512 

10-279831 

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9-947136 

18 

43 

9-667546 

401 

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511 

10-279524 

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111 

9-947070 

17 

44 

9-667786 

401 

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9-720783 

511 

10-279217 

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9-947004 

16 

45 

9-668027 

400 

•331973 

9-721089 

511 

10-278911 

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111 

9-946937 

15 

46 

9-668267 

400 

•331733 

9-721396 

511 

10-278604 

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111 

9-946871 

14 

47 

9-668506 

400 

•331494 

9-721702 

511 

10-278298 

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111 

9-946804 

13 

48 

9-668746 

399 

•331254 

9-722009 

510 

10-277991 

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111 

9-946738 

12 

49 

9-668986 

399 

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9-722315 

510 

10-277685 

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111 

9-946671 

11 

50 

9-669225 

399 

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9-722621 

510 

10-277379 

•053396 

111 

9-946604 

10 

51 

9-669464 

399 

•330536 

9-722927 

510 

10-277073 

•053462 

111 

9-946538 

9 

52 

9-669703 

398 

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9-723232 

510 

10-276768 

•053529 

111 

9-946471 

8 

53 

9-669942 

398 

•330058 

9-723538 

509  110-276462 

•053596 

111 

9-946404 

7 

54 

9-670181 

398 

•329819 

9-723844 

509 

10-276156 

•053663 

111 

9-946337 

6 

55 

9-670419 

397 

•329581 

9-724149 

509 

10-275851 

•053730 

1119-946270 

5 

56 

9-670658 

397 

•329342 

9-724454 

509 

10-275546 

•053797 

1129  -946203 

4 

57 

9-670896 

397 

•329104 

9-724759 

509 

10-275241 

•053804 

11219-946136 

3 

58 

9-671134 

397  -328866 

9-725065 

508 

10-274935 

•053931J112I9-946069 

9 

59 

9-671372 

396  i  -328628 

9-725369J  508 

10-274631 

•053998 

112 

9-946002 

1 

60 

9-671609 

396  -328391 

9-725674  508  10-274326 

•054065 

112 

9-945935 

0 

' 

Cosine. 

Secant.   li  Cotangent.      !   Tangent.  |!  Cosecant. 

Sine. 

' 

62  DKG. 


LOGARITHMIC   SINES,  ETC, 


60 


' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

i 

0 

9-671609 

•328391 

9-725674 

10-274326 

•054065 

9-945935 

60 

1 

9-671847 

396 

•328153 

9-725979 

508 

10-274021 

•054132 

112 

9-945868 

59 

2 

9-672084 

395 

•327916 

9-726284 

508 

10-273716 

•054200 

112 

9-945800 

58 

3 

9-672321 

395 

•327679 

9-726588 

507 

10-273412 

•054267 

112 

9-945733 

57 

4 

9-672558 

395 

•327442 

9-726892 

507 

10-273108 

•054334 

112 

9-945666 

56 

5 

9-672795 

395 

•327205 

9-727197 

507 

10-272803 

•054402 

112 

9-945598 

55 

6 

9-673032 

394 

•326968 

9-727501 

507 

10-272499 

•054469 

112 

9-945531 

54 

7 

9-673268 

394 

•326732 

9-727805 

507 

10-272195 

•054536 

112 

9-945464 

53 

8 

9-673505 

394 

•326495 

9-728109 

506 

10-271891  11-054604 

113 

9-945396 

52 

9 

9-673741 

394 

•326259 

9-728412 

506 

10-271588 

•054672  113 

9-945328 

51 

10 

9-673977 

393 

•326023 

9-728716 

506 

10-271284 

•054739 

113 

9-945261 

50 

11 

9-674213 

393 

•325787 

9-729020 

506 

10-270980 

•054807 

113 

9-945193 

49 

12 

9-674448 

393 

-325552 

9-729323 

506 

10-270677 

•054875 

113 

9-945125 

48 

13 

9-674684 

392 

•325316 

9-729626 

505 

10-270374 

•054942 

113 

9-945058 

47 

14 

9-674919 

392 

•325081 

9-729929 

505 

10-270071  ij-055010 

113 

9-944990 

46 

15 

9-675155 

392 

•324845 

9-730233 

505 

10-269767 

•055078 

113 

9-944922 

45 

16 

9-675390 

392 

•324610 

9-730535 

505 

10-269465 

•055146 

113 

9-944854 

44 

17 

9-675624 

391 

•324376 

9-730838 

505 

10-269162 

•055214 

113 

9-944786 

43 

18 

9-675859- 

391 

•324141 

9-731141 

504 

10-268859 

•055282)113 

9-9447,18 

42 

19 

9-676094 

391 

•323906 

9-731444 

504 

10-268556 

•055350 

113 

9-944650 

41 

20 

9-676328 

391 

•323672 

9-731746 

504 

10-268254 

•055418 

113 

9-944582 

40 

21 

9-676562 

390 

-  -323438 

9-732048 

504 

10-267952 

•055486 

114 

9-944514 

39 

22 

9-676796 

390 

•323204 

9-732351 

504 

10-267649 

•055554 

114 

9-944446 

38 

23 

9-677030 

390 

•322970 

9-732653 

503 

10-267347 

•055623 

114 

9-944377 

37 

24 

9-677264 

390 

•322736 

9-732955 

503 

10-267045 

•055691 

114 

9-944309 

36 

25 

9-677498 

389 

•322502 

9-733257 

503 

10-266743 

•055759 

114 

9-944241 

35 

26 

9-677731 

389 

•322269 

9-733558 

503 

10-266442 

•055828 

114 

9-944172 

34 

27 

9-677964 

389 

•322036 

9-733860 

503 

10-266140 

•055896 

114 

9-944104  33 

28 

9-678197 

388 

•321803 

9-734162 

503 

10-265838 

•055964 

114 

9-944036 

32 

29 

9-678430 

388 

•321570 

9-734463 

502 

10-265537 

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114 

9-943967 

31 

30 

9-678663 

388 

•321337 

9-734764 

502 

10-265236 

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114 

9-943899 

30 

31 

9-678895 

388 

•321105 

9-735066 

502 

10-264934 

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114 

9-943830 

29 

32 

9-679128 

387 

•320872 

9-735367 

502 

10-264633 

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114 

9-943761 

28 

33 

9-679360 

387 

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9-735668 

502 

10-264332 

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114 

9-943693 

27 

34 

9-679592 

387 

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9-735969 

501 

10-264031 

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115 

9-943624 

26 

35 

9-679824 

387 

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9-736269 

501 

10-263731 

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115 

9-943555 

25 

36 

9-680056 

386 

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9-736570 

501 

10-263430 

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115 

9-943486 

24 

37 

9-680288 

386 

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9-736871 

501 

10-263129 

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115 

9-943417 

23 

38 

9-680519 

386 

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9-737171 

501 

10-262829 

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115 

9-943348 

22 

39 

9-680750 

385 

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9-737471 

500 

10-262529 

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115 

9-943279 

21 

40 

9-680982 

385 

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9-737771 

500 

10-262229 

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115 

9-943210 

20 

41 

9-681213 

385 

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9-738071 

500 

10-261929 

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115 

9-943141 

19 

42 

9-681443 

385 

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500 

10-261629 

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115 

9-943072 

18 

43 

9-681674 

384 

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10-261329 

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115 

9-943003 

17 

44 

9-681905 

384 

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10-261029 

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115 

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9-682135 

384 

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10-260729 

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9-942864 

15 

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384 

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10-260430 

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115 

9-942795 

14 

47 

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383 

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10-260130 

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116 

9-942726 

13 

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9-682825 

383 

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9-740169 

499 

10-259831 

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116 

9-942656 

12 

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9-683055 

383 

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10-259532 

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116 

9-942587 

11 

50 

9-683284 

383 

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498 

10-259233 

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116 

9-942517 

10 

51 

9-683514 

382 

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498 

10-258934 

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116 

9-942448 

9 

52 

9-683743 

382 

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9-741365 

498 

10-258635 

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116 

9-942378 

8 

53 

9-683972 

382 

•316028 

9-741664 

498 

10-258336 

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116 

9-942308 

7 

54 

9-684201 

382 

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9-741962 

498 

10-258038 

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116 

9-942239 

6 

55 

9-684430 

381 

•315570 

9-742261 

498 

10-257739 

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116 

9-942169 

5 

56 

9-684658 

381 

•315342 

9-742559 

497 

10-257441 

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116 

9-942099 

4 

57 

9-684887 

381 

•315113 

9-742858 

497 

10-257142 

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116 

9-942029 

3 

58 

9-685115 

380 

•314885 

19-743156 

497 

10-256844 

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116 

9-941959 

2 

59 

9-685343 

380 

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19-743454 

497 

10-256546 

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116 

9-941889 

1 

60 

9-685571 

380 

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1  9-743752 

497 

10-256248 

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117 

9-941819 

0 

'  •   Cosine. 

Secant.  ''Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

61  DEG. 


70 


LOGARITHMIC    SINES,  ETC. 


29  BEG. 


1 

Sine. 

?$>   Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Difl 
100' 

Cosine. 

0 

9-685571 

•314429 

9-743752 

10-256248 

•058181 

9-941819 

60 

1 

9-685799 

380  -314201 

9-744050  496 

10-255950 

•058251 

1179-941749 

59 

2 

9  '686027 

379  -313973 

-9-744348J  496 

10-255652 

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117 

9-941679 

58 

3f|  9-686254 

379  :  -313746 

9-744645  496 

10-255355 

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117 

9-941609 

57 

4  9-686482 

379  -313518 

9-744943 

496 

10-255057 

•058461 

117  9-941539 

56 

5  ;  9-686709 

379  !  -313291 

9-745240 

496 

10-254760 

•058531 

117  9-941469 

55 

6  9-686936 

378  -313064 

9-745538 

496 

10-254462 

•058602 

117I9-941398 

54 

7  ;  9-687163 

378  i  -312837  i  9-745835 

495 

10-254165 

•058672 

1179-941328 

53 

8  19-687389 

378  -312611  1  9-746132 

495 

10-253868 

•058742 

11719-941258 

52 

9  |i  9-687616 

378  -312384  9-746429 

495 

10-253571 

•058813 

1179-941187 

51 

10  1  9-687843 

377  |  -312157  ''•  9-746726 

495 

10-253274 

•058883  1179-941117 

50 

11  9-688069 

377 

•311931  9-747023 

495 

10-252977 

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49 

12  !l  9-688295 

377 

•311705  '  9-747319 

494 

10-252681 

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48 

13  j  9-688521 

377 

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10-252384 

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10-252087 

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10-251791 

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9-689198 

376 

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10-251495 

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9-689423 

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10-251199 

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118 

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375 

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10-250903 

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118 

9-940551 

42 

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375 

•310127  :!  9-749393 

493 

10-250607  (-059520 

118 

9-940480 

41 

20 

9-690098 

375 

•309902  ! 

9-749689 

493 

10-250311  [1-059591 

118 

9-940409 

40 

21 

9-690323 

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9-749985 

493 

10-250015 

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118 

9-940338 

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22 

i  9-690548 

374 

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9-750281 

493 

10-249719 

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9-940267 

38 

23  9-690772 
24  1|  9-690996 

374 
374 

•309228  i  9-750576 
•309004  !!  9-750872 

493 
492 

10-249424 
10-249128 

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9-940196 
9-940125 

37 
36 

25  j  9-691220 

374 

•308780  19-751167 

492 

10-248833 

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119 

9-940054 

35 

26 

9-691444 

373 

•308556  19-751462 

492 

10-248538 

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119 

9-939982 

34 

27!i9-691668 

373 

•308332  9-751757 

492 

10-248243 

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119 

9-939911 

33 

28  i  9-691892 

373 

•308108  9-752052 

492 

10-247948 

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119 

9-939840 

32 

29119-692115 

373 

•307885 

9-752347 

491 

10-247653 

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119 

9-939768 

31 

30  1  9-692339 

372 

•307661 

9-752642 

491 

10-247358 

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119 

9-939697 

30 

31  i  9-692562 

372 

•307438 

9-752937 

491 

10-247063 

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119 

9-939625 

29 

32  ||  9-692785 

372 

•307215 

9-753231 

491 

10-246769 

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119 

9-939554 

28 

33  i  9-693008  371 

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9-753526 

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10-246474 

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9-939482 

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34  0-693231'  371 

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10-246180 

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9-939410 

26 

35 

9-693453  371 

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9-754115 

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10-245885 

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119 

9-939339 

25 

36 

9-693676:  371 

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9-754409 

490 

10-245591 

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119 

9-939267 

24 

37  19-693898'  370 

•306102 

9-754703 

490 

10-245297 

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9-939195 

23 

38 

9-694120  370 

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9-754997 

490 

10-245003 

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9-939123 

22 

39 

9-694342!  370 

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9-755291 

490 

10-244709 

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9-939052 

21 

40  j|  9-694564  s  370 

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9-755585 

490 

10-244415 

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9-938980 

20 

41  i  9-694786  369 

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9-755878 

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10-244122 

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9-938908 

19 

42  j  9-6950071  369 

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9-756172 

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9-938836 

18 

43 

9-695229!  369 

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9-756465 

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10-243535 

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120 

9-938763 

17 

44  9-6954501  369 

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10-243241 

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9-938691 

16 

45  !  9-695671  368 

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10-242948 

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9-938619 

15 

46 

9-695892!  368 

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9-757345 

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10-242655 

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9-938547 

14 

47 

9-6961131  368 

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9-938475 

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49  j  9-696554  367 

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50  jl  9-696775;  367 

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51  i  9-6969951  367 

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52  19-697215  367 

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9-938113 

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58  9-697486  366 

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9-938040 

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54  •  9-697654'  366 

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55  1  9  -697874  :  366 

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56  Ij  9  -698094  [  366 

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9-937822 

4 

57  i  9-698313  365 

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10-239436 

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9-937749 

3 

58  i  9-698532  365 

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9-937676 

2 

5yl!9-t>i)8751  365 

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10-233852 

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121 

9-937604 

1 

W  :  9-698970  365 

•301030 

9-761439 

486 

10-238561 

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121 

9-937531 

0 

Cosiue.   ; 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine.   1 

' 

60  UKG. 


LOGARITHMIC    SINES,  ETC. 


71 


30  PEG. 


• 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

i 

~~0 

9-698970 

•301030 

;  9-761439 

10-238561 

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9-937531 

60~ 

1 

9-699189 

364 

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486 

10-238269 

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9-937458 

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2 

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3 

9-699626 

364 

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10-237686 

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9-937312 

57 

4 

9-699844 

364 

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9-762606 

486 

10-237394 

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122 

9-937238  50 

5 

9-700062 

363 

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9-762897 

485 

10-237103 

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122 

9-937165  55 

0 

9-700280 

363 

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9-763188 

485 

10-236812 

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122 

9-937092JI54 

7 

9-700498 

363  -299702 

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485 

10-236521 

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122  9-937019 

53 

8 

9-700716 

363  -299284 

9-763770 

485 

10-236230 

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9-936946 

52 

9 

9-700933 

363 

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9-764061 

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10-235939 

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9-936872 

62 

10 

9-701151 

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10-235648 

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122 

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9-701368 

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485 

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12 

9-701585 

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13 

9-701802 

362 

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9-936578 

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14 

9-702019 

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123  9-936505 

46 

15 

9-702236 

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484 

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16 

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9-936357 

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17 

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9-935914 

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23 

9-703964 

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10-231876 

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9-935840 

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24 

9-704179 

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9-935766 

36 

25 

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9-935692 

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26 

9-704610 

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28 

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9-934948!l25 

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9-934873 

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9-934798  23 

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9-934274  16 

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9-709306 
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52 

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53 

9-710364 

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54 

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352 

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10-222945 

066480126|9-933520  !  6 

55 

9-710786 

352 

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10-222658  i 

066555  '1269-933445  5 

56 

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9-777628  478  10-222372  ! 

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9-933369  4 

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9-933293  3 

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59 

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60 

9-711839 

350 

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9-778774  477  10-221226  -066.  >o4  120:9-933066  0 

1 

'  [   Cosine. 

i   Secant. 

i  Cotangent.         T:KI-VU:.    Coscviint.        Sine.   1;  ' 

G2 


59  u  KG. 


72 


LOGARITHMIC    SINES,  ETC. 


31  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 

100" 

Cosine. 

' 

0 

9-711839 

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9-778774 

10-221226 

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9-933066 

60 

1 

9-712050 

350 

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10-220940 

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126 

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2 

9-712260 

350 

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3 

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350 

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4 

9-712679 

349 

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127 

9-932762:56 

5 

9-712889 

349 

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9-780203 

476 

10-219797 

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127 

9-932685  55 

6 

9-713098 

349 

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476 

10-219511 

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127 

9-932609  54 

7 

9-713308 

349 

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9-780775 

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127 

9-932533J53 

8 

9-713517 

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127 

9-932457!  52 

9 

9-713726 

348 

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9-781346 

476 

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127 

9-932880!  51 

10 

9-713935 

348 

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9-781631 

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127 

9-932304!;50 

11 

9-714144 

348 

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9-781916 

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127 

9-932228:|49 

12 

9-714352 

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9-9321  51JJ48 

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128 

9-931998|46 

15 

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128 

9-931921  45 

16 

9-715186 

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128 

9-931845!  44 

17 

9-715394 

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9-931768  43 

18 

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9-931691  42 

19 

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9-931537  40 

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24 

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9-931229  36 

25 

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10-214100 

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129  9-931152!!35 

26 

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10-213816 

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9-929989 

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•071028 

131 

9-928972 

7 

54 

9-722994 

338 

•277006  !  9-794101 

469 

10-205899 

•071107 

131 

9-928893 

6 

55 

9-723197 

338 

•276803  9-794383 

469 

10-205617 

•071185 

131 

9-928815 

5 

56 

9-723400 

338 

•276600  9-794664 

469 

10-205336 

•071264 

131 

9-928736 

4 

57 

9-723603 

338 

•276397  :  9-794945 

469 

10-205055 

•071343 

131 

9-928657! 

3 

58 

9-723805 

337 

•276195  9-795227 

469 

10-204773 

•071422 

131 

9-928578 

2 

59 

9-724007 

337 

•275993  !  9-795508 

469 

10-204492 

•071501 

131 

9-928499 

1 

60 

9-724210 

337 

•275790  9-795789 

468 

10-204211 

•071580 

131 

9-928420: 

0 

' 

Cosine. 

Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine.   ||' 

58  DEG. 


LOGARITHMIC    SINES,  ETC 


32  DEG, 


/ 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

' 

0 

9-724210 

•275790 

9-795789 

10  -2042  11 

•071580 

9-928420 

60 

1 

9-724412 

337 

•275588 

9-796070 

468 

10-203930 

•071658 

132 

9-928342 

59 

2 

9-724614 

337 

•275386 

9-796351 

468 

10-203649 

•071737 

132 

9-928263 

58 

3 

9-724816 

336 

•275184 

9-796632 

468 

10-203368 

•071817 

132 

9-928183 

57 

4 

9-725017 

336 

•274983 

9-796913 

468 

10-203087 

•071896 

132 

9-928104  !56 

5 

9-725219 

336 

•274781 

9-797194 

468 

10-202806 

•071975 

132 

9-928025  55 

6 

9-725420 

336 

•274580 

9-797475 

468 

10-202525 

•072054 

132 

9-927946;  54 

7 

9-725622 

335 

•274378 

9-797755 

468 

10-202245 

•072133 

132 

9-927867  53 

8 

9-725823 

335 

•274177 

9-798036 

467 

10-201964 

•072213 

132 

9-927787I52 

9 

9-726024 

335 

•273976 

9-798316 

467 

10-201684 

•072292 

132 

9-927708  51 

10 

9-726225 

335 

•273775 

9-798596 

467 

10-201404 

•072371 

132 

9-927629|50 

11 

9-726426 

335 

•273574 

9-798877 

467 

10-201123 

•072451 

132 

9-927549:49 

12 

9-726626 

334 

•273374 

9-799157 

467 

10-200843 

•072530 

132 

9-927470!  48 

13 

9-726827 

334 

•273173 

9-799437 

467 

10-200563 

•072610 

133 

9-927390  47 

14 

9-727027 

334 

•272973 

9-799717 

467 

10-200283 

•072690 

133 

9-927310  46 

15 

9-727228 

334 

•272772 

9-799997 

467 

10-200003 

•072769 

133 

9-927231  45 

16 

9-727428 

334 

•272572 

9-800277 

466 

10-199723 

•072849 

133 

9-927151!  44 

17 

9-727628 

333 

•272372 

9-800557 

466 

10-199443 

•072929 

133 

9-927071'  43 

18 

9-727828 

333 

•272172 

9-800836 

466 

10-199164 

•073009 

133 

9-926991  42 

19 

9-728027 

333 

•271973 

9-801116 

466 

10-198884 

•073089  133 

9-926911  41 

20 

9-728227 

333 

•271773 

9-8013961  466 

10-198604 

•073169 

133 

9-926831  40 

21 

9-728427 

333 

•271573 

9-801675 

466 

10-198325 

•073249 

133 

9-926751  39 

22 

9-728626 

332 

•271374 

9-801955 

466 

10-198045 

•073329 

133 

9-926671  '38 

23 

9-728825 

332 

•271175 

9-802234 

466 

10-197766 

•073409 

133 

9  -926591  '37 

24 

9-729024 

332 

•270976 

9-802513 

465 

10-197487 

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133 

9-926511  36 

25 

9-729223 

332 

•270777 

9-802792 

465 

10-197208 

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134 

9-926431  35 

26 

9-729422 

331 

•270578 

9-803072 

465 

10-196928 

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134 

9-926351  34 

27 

9-729621 

331 

•270379 

9-803351 

465 

10-196649 

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134 

9-926270  33 

28 

9-729820 

331 

•270180 

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465 

10-196370 

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134 

9-926190  32 

29 

9-730018 

331 

•269982 

9-803908 

465 

10-196092 

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134 

9-926110  31 

30 

9-730217 

330 

•269783 

9-804187 

465 

10-195813 

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134 

9-926029 

30 

31 

9-730415 

330 

•269585 

9-804466 

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10-195534 

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134 

9-925949 

29 

32 

9-730613 

330 

•269387 

9-804745 

464 

10-195255 

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9-925868 

28 

33 

9-730811 

330 

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9-805023 

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10-194977 

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134 

9-925788 

27 

34 

9-731009 

330 

•268991 

9-805302 

464 

10-194698 

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134 

9-925707 

26 

35 

9-731206 

329 

•268794 

9-805580 

464 

10-194420 

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134 

9-925626 

25 

36 

9-731404 

329 

•268596 

9-805859 

464 

10-194141 

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134 

9-925545 

24 

37 

9-731602 

329 

•268398 

9-806137 

464 

10-193863 

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135 

9-925465 

23 

38 

9-731799 

329 

•268201 

9-806415 

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10-193585 

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135 

9-925384 

22 

39 

9-731996 

329 

•268004 

9-806693 

464  J10-193307 

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135 

9-925303 

21 

40 

9-732193 

328 

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9-806971 

463  10-193029 

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135 

9-925222 

20  ! 

41 

9-732390 

328 

•267610 

9-807249 

463  10-192751 

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135 

9-925141 

19 

42 

9-732587 

328 

•267413 

9-807527 

463  10-192473 

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135 

"9-925060 

18 

43 

9-732784 

328 

•267216 

9-807805 

463  10-192195 

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135 

9-924979 

17 

44 

9-732980 

328 

•267020 

9-808083 

463 

10-191917 

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135 

9-924897 

16 

45 

9-733177 

327 

•266823 

9-808361 

463 

10-191639 

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135 

9-924816 

16 

46 

9-733373 

327 

•266627 

9-808638 

463  10-191362 

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135 

9-924735 

14 

47 

9-733569 

327 

•266431 

9-808916 

463  10-191084 

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136 

9-924654 

13 

48 

9-733765 

327 

•266235 

9-809193 

462  10-190807 

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136 

9-924572 

12 

49 

9-733961 

327 

•266039 

9-809471 

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9-924491 

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5€ 

9-734157 

326 

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9-809748 

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9-924409 

10 

51 

9-734353 

326 

•265647 

9-810025 

462  110-189975 

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136 

9-924328 

9 

52 

9-734549 

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•265451 

9-810302 

462  10-189698 

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186 

9-924246 

8 

53 

9-734744 

326 

•265256 

9-810580 

462  10-189420 

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136 

9-924164 

7 

54 

9-734939 

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•265061 

9-810857 

462 

10-189143 

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136 

9-924083 

6 

55 

9-735135 

325 

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10-188866 

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136 

9-924001 

5 

56 

9-735330 

325 

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9-811410 

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10-188590 

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136 

9-923919 

4 

57 

9-735525 

325 

•264475 

9-811687 

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10-188313 

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136 

9-923837 

3 

58 

9-735719 

325  -264281 

9-811964 

461 

10-188036 

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136 

9-923755 

2 

59 

9-735914 

324  -264086  !  9-812241 

461 

10-187759 

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137 

9-923673 

1 

60 

9-736109 

324  -2638911  9-812517 

461 

10-187483 

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137 

9-923591 

0 

i 

Cosine. 

Secant.    Cotangent.  !        Tangenc.  i  Cosecant.) 

Sine. 

' 

57  DKG. 


74 


LOGARITHMIC    SINES,  ETC. 


r 

Sine. 

jjjy)      Cosecant,    i     Tangent. 

Diff. 
100" 

Cotangent. 

Secant.     J^ff 

1     Cosine.     ||    ' 

0 

9-736109 

•263891    9-812517 

10-187483 

•076409' 

'9-9235911160 

•t 

9-736303 

324;  -263697    9-812794 

461 

10-187206 

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1  37  ;9  -923509  59 

2 

9-736498 

324:  -263502  I1  9-813070 

461 

10-186930 

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19-923427  58 

3 

9-736692 

324  |  -263308  ;!  9-813347 

461 

10-186653 

•076655  13719-923345  57 

4 

9-736886 

323  !  -263114:1  9-813623 

460 

10-186377 

•076737I137J9-923263I56 

5 

9-737080 

323  !  -262920  !  9-818899 

460 

10-186101 

•0768191137 

19-923181  55 

6 

9-737274 

323     -262726  19-814175 

460 

10-185825 

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9-923098  54 

7  |  9-737467 

323     -262533    9-814452 

460 

10-185548 

•076984  137 

9-923016  153 

8    9-737661 

323 

•262339  !  9-814728 

460 

10-185272 

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9-922933  Io2 

9    9-737855 

322 

•2(52145  '9-815004 

460 

10-184996 

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9-922851  51 

10  II  9-738048 

322 

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460 

10-184721 

•077232|l37  9-922768150 

11 

9-738241 

322 

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460 

10-184445 

•077314  138i9-922686  49 

12 

9-738434 

322 

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460 

10-184169 

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9-922603  48 

13 

9-738627 

322 

•261373    9-816107 

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10-183893 

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14 

9-738820 

321 

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459 

10-183618 

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15 
16 

9-739013 
9-739206 

321 
321 

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10-183342 
10-183067 

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17 

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143 

18 

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19 

9-739783 

320 

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10-182241 

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20 

9-739975 

320     -260025  II  9-81  8086 

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10-181965 

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21 

9-740167 

320 

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10-181690 

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9-921857  139 

22 

9-740359 

320 

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10-181415 

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9-921774j!38 

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9-740550 

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10-181140 

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9-921691  37 

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9-740934 

319 
319 

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10-180865 
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9  -921607|  36 
9-921524  35 

26 

9-741125 

319 

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10-180316 

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9-921441!  34 

27 

9-741316 

319 

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10-180041 

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28 

9-741508 

319 

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9-741699 

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33  i!  9-742462  318 

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9-920856 

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9-742652  317 

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9-8218801  457  10-178120 

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9-920772  26 

35 

9-742842  317 

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9-920688:25 

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9-743033  317 

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9-92060424 

37 

9-743223  317 

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9-920520  23 

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9-743413  317 

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39 

9-743602  316 

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40 

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41 

9-743982  316 

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50 

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11 
10 

51 
52 

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53 

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57  19-746999  313 

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10-171830 

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3 

58    9-747187  313 

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10-171558 

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2 

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0 

'     ;'       Cosiue.                       Secant.      :i  Cotangent.  '            '      Tangent.       'Cosecant. 

Sine.       I! 

56  DEC. 


LOGARITHMIC    SINES,  ETC 


75 


34  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100'' 

Cosine. 

' 

~~0 

9-747562 

•252438 

9-828987 

10-171013 

•081426 

9-918574 

60~ 

1 

9-747749 

312 

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2 

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9-918404 

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3 

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312 

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10-170195 

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142 

9-918318 

57 

4 

9-748310 

311 

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454 

10-169923 

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142 

9-918233 

56 

5 

9-748497 

311 

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454 

10-169651 

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142 

9-918147 

55 

6 

9-748683 

311 

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10-169379 

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142 

9-918062 

54 

7' 

9-748870 

311 

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453 

10-169107 

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143 

9-917976 

53 

8 

9-749056 

311 

•250944 

9-831165 

453 

10-168835 

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143 

9-917891 

52 

9 

9-749243 

310 

•250757 

9-831437 

453 

10-168563 

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143 

9-917805 

51 

10 

9-749429 

310 

•250571 

9-831709 

453 

10-168291 

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143 

9-917719 

50 

11 

9-749615 

310 

•250385 

9-831981 

453 

10-168019 

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143 

9-917634 

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12 

9-749801 

310 

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10-167747 

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143 

9-917548 

48 

13 

9-749987 

310 

•250013 

9-832525 

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10-167475 

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143 

9-917462 

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14 

9-750172 

309 

•249828 

9-832796 

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10-167204 

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143 

9-917376 

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15 

9-750358 

309 

•249642 

9-833068 

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10-166932 

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143 

9-917290 

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16 

9-750543!  309 

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10-166661 

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9-917204 

44 

17 

9-750729 

309 

•249271 

9-833611 

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10-166389 

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143 

9-917118 

43 

18 

9-750914 

309 

•249086 

9-833882 

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10-166118 

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9-917032 

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19 

9-751099 

308 

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9-834154 

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10-165846 

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144 

9-916946 

41 

20 

9-751284 

308 

•248716 

9-834425 

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10-165575 

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9-916859 

40 

21 

9-751469 

308 

•248531 

9-834696 

452 

10-165304 

•083227 

144 

9-916773 

39 

22 

9-751654!  308 

•248346 

9-834967 

452 

10-165033 

•083313 

144 

9-916687 

38 

23 

9-751839  308 

•248161 

9-835238 

452 

10-164762 

•083400 

144 

9-916600 

37 

24 

9-752023 

308 

•247977 

9-835509 

452 

10-164491 

•083486 

144 

9-916514 

36 

25 

9-752208 

307 

•247792 

9-835780 

452 

10-164220 

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144 

9-916427 

35 

26 

9-752392  307 

•247608 

9-836051 

452 

10-163949 

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144 

9-916341 

34 

27 

9-752576  307 

•247424 

9-836322 

451 

10-163678 

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144 

9-916254 

33 

28 

9-752760  307 

•247240 

9-836593 

451 

10-163407 

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144 

9-916167 

32 

29 

9-752944  307 

•247056 

9-836864 

451 

10-163136 

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145 

9-916081 

31 

30 

9-753128  306 

•246872 

9-837134 

451 

10-162866 

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9-915994 

30 

31 

9-753312  306 

•246688 

9-837405 

451 

10-162595 

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145 

9-915907 

29 

32 

9-753495 

306 

•246505 

9-837675 

451 

10-162325 

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145 

9-915820 

28 

33 

9-753679 

306 

•246321 

9-837946 

451 

10-162054 

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145 

9-915733 

27 

34 

9-753862 

306 

•246138 

9-838216 

451 

10-161784 

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145 

9-915646 

26 

35 

9-754046 

305 

•245954 

9-838487 

451 

10-161513 

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145 

9-915559 

25 

36 

9-754229 

305 

•245771 

9-838757 

451 

10-161243 

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145 

9-915472 

24 

37 

9-754412 

305 

•245588 

9-839027 

450 

10-160973 

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145 

9-915385 

23 

38 

9-754595 

305 

•245405 

9-839297 

450 

10-160703 

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145 

9-915297 

22 

39 

9-754778 

305 

•245222 

9-839568 

450 

10-160432 

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145 

9-915210 

21 

40 

9-754960 

304 

•245040 

9-839838 

450 

10-160162 

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145 

9-91of23 

20 

41 

9-755143 

304 

•244857 

9-840108 

450 

10-159892 

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146 

9-915035 

19 

42 

9-755326 

304 

•244674 

9-840378 

450 

10-159622 

•085052 

146 

9-914948 

18 

43 

9-755508 

304 

•244492 

9-840647 

450 

10-159353 

•085140 

146 

9-914860 

17 

44 

9-755690 

304 

•244310 

9-840917 

450 

10-159083 

•085227 

146 

9-914773 

16 

45 

9-755872 

304 

•244128 

9-841187 

450 

10-158813 

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146 

9-914685 

15 

46 

9-756054 

303 

•243946 

9-841457 

450 

10-158543 

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146 

9-914598 

14 

47 

9-756236 

303 

•243764 

9-841726 

449 

10-158274 

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146 

9-914510 

13 

48 

9-756418 

303 

•243582 

9-841996 

449 

10-158004 

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146 

9-914422 

12 

49 

9-756600 

303 

•243400 

9-842266 

449 

10-157734 

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146 

9-914334 

11 

50 

9-756782 

303 

•243218 

9-842535 

449 

10-157465 

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146 

9-914246 

10 

51 

9-756963 

302 

•243037 

9-842805 

449 

10-157195 

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147 

9-914158 

9 

52 

9-757144 

302 

•242856 

9-843074 

449 

10-156926 

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147 

9-914070 

8 

53 

9-757326 

302 

•242674 

9-843343 

449 

10-156657 

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147 

9-913982 

7 

54 

9-757507 

302 

•242493 

9-843612 

449 

10-156388 

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147 

9-913894 

6 

55 

9-757688 

302 

•242312 

9-843882 

449 

10-156118 

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147 

9-913806 

5 

56 

9-757869 

301 

•242131 

9-844151 

449 

10-155849 

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147 

9-913718 

4 

57 

9-758050 

301 

•241950 

9-844420 

448 

10-155580 

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147 

9-913630 

3 

58 

9-758230 

301 

•241770 

9-844689 

448 

10-155311 

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147 

9-913541 

2 

59 

<)  -758411 

301 

•241589 

9-844958 

448 

10-155042 

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147 

9-913453 

1 

60 

9-758591 

301 

•241409 

9-845227 

448 

10-154773 

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147 

9-913365 

0 

'     Cosine. 

I   Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

55  DEC. 


76 


LOGARITHMIC    SINES,  ETC. 


35  DEG. 


' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 

100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

> 

0 

9-758591 

•241409 

9-845227 

10-154773 

•086635 

9-913365 

60 

1 

9-758772 

301 

•241228 

9-845496 

448 

10-154504 

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147 

9-913276 

59 

2 

9-758952 

300 

•241048 

9-845764 

448 

10-154236 

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147 

9-913187 

58 

3 

9-759132 

300 

•240868 

9-846033 

448 

10-153967 

•086901 

148 

9-913099 

57 

4 

9-759312 

300 

•240688 

9-846302 

448 

10-153698 

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148 

9-913010 

56 

5 

9-759492 

300 

•240508 

9-846570 

448 

10-153430 

•087078 

148 

9-912922 

55 

6 

9-759672 

300 

•240328 

9-846839 

448 

10-153161 

•087167 

148 

9-912833 

54 

7 

9-759852 

299 

•240148 

9-847107 

448 

10-152893 

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148 

9-912744 

53 

8 

9-760031 

299 

•239969 

9-847376 

447 

10-152624 

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148 

9-912655 

52 

9 

9-760211 

299 

•239789 

9-847644 

447 

10-152356 

•087434 

148 

9-912566  51 

10 

9-760390 

299 

•239610 

9-847913 

447 

10-152087 

•087523 

148 

9-912477  50 

11 

9-760569 

299 

•239431 

9-848181 

447 

10-151819 

•087612 

148 

9-912388 

49 

12 

9-760748 

298 

•239252 

9-848449 

447 

10-151551 

•087701 

148 

9-912299 

48 

13 

9-760927 

298 

•239073 

9-848717 

447  10-151283 

•087790 

149 

9-912210 

47 

14 

9-761106 

298 

•238894 

9-848986 

447 

10-151014 

•087879 

149 

9-912121 

46 

15 

9-761285 

298 

•238715 

9-849254 

447 

10-150746 

•087969 

149 

9-912031 

45 

16 

9-761464 

298 

•238536 

9-849522 

447 

10-150478 

•088058 

149 

9-911942 

44 

17 

9-761642 

298 

•238358 

9-849790 

447  10-150210 

•088147 

149 

9-911853 

43 

18 

9-761821 

297 

•238179  9-850058 

446  10-149942 

•088237 

149 

9-911763 

42 

19 

9-761999 

297 

•238001  9-850325 

446  10-149675 

•088326 

149 

9-911674 

41 

20 

9-762177 

297 

•237823  19-850593 

446  10-149407 

•088416 

149 

9-911584 

40 

21 

9-762356 

297 

•237644  9-850861 

446  10-149139 

•088505 

149 

9-911495 

39 

22 
23 
24 

9-762534 
9-762712 
9-762889 

297 
296 
296 

•237466  1|  9-851129 
•237288  ||9-851396 
•237111'  9-851664 

446  110-148871 
446  ilO-148604 
446  10-148336 

088595 
088685 
088774 

149 
149 
150 

9-911405 
9-911315 
9-911226 

38 
37 
36 

25 

9-763067 

296 

•236933!  9-851931 

446  10-148069 

088864 

150 

9-911136 

35 

26 

9-763245 

296 

•236755  9-852199 

446  10-147801 

088954 

150 

9-911046 

34 

27 

9-763422 

296 

•236578  9-852466 

446  10-147534 

089044 

150 

9-910956  '33 

28 

9-763600 

296 

•2364001  9-852733 

446  10-147267 

089134 

150 

9-910866  J32 

29 

9-763777 

295 

•236223!  9-853001 

445 

10-146999 

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150 

9-910776  81 

30 

9-763954 

295 

•236046  9-853268 

445 

10-146732 

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150 

9-910686!30 

31 

9-764131 

295 

•235869  i  9-853535 

445 

10-146465 

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150 

9-910596:129 

32 

9-764308 

295 

•235692  9-853802 

445 

10-146198 

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150 

9-910506  28 

33 

9-764485 

295 

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445 

10-145931 

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150 

9-910415  27 

34 

9-764662 

294 

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445 

10-145664 

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150 

9-910325J26 

35 

9-764838 

294 

•235162!  9-854603 

445 

10-145397 

089765 

151 

9-910235  25 

36 

9-765015 

294 

•234985  9-854870 

445 

10-145130 

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151 

9-910144124 

37 

9-765191 

294 

•234809!!  9-855137 

445 

10-144863 

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151 

9-910054  !23 

38 

9-765367 

294 

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445 

10-144596 

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151 

9-909963J22 

39 

9-765544 

294 

•234456  9-855671 

445 

10-144329 

090127 

151 

9-909873  21 

40 

9-765720 

293 

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444 

10-144062 

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151 

9-909782  20 

41 

9-765896 

293 

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10-143796 

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151 

9-90969119 

42 

9-766072 

293 

•233928 

9-856471 

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10-143529 

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151 

9-909601  18 

43 

9-766247 

293 

•233753 

9-856737 

444 

10-143263 

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151 

9-909510  17 

44 

9-766423 

293 

•233577 

9-857004 

444 

10-142996 

090581 

151 

9-909419  16 

45 

9-766598 

293 

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9-857270 

444 

10-142730 

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151 

9-909328JI15 

46 

9-766774 

292 

•233226 

9-857537 

444 

10-142463 

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152 

9*900287|14 

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9-766949 

292 

•233051 

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10-142197 

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152 

9-909146 

13 

48 

9-767124 

292 

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9-858069 

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10-141931 

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152 

9-909055 

12 

49 

9-767300 

292 

•232700 

9-858336 

444 

10-141664 

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152 

9-908964 

11 

50 

9-767475 

292 

•232525 

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444 

10-141398 

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152 

9-908873! 

10 

51 

9-767649 

291 

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444 

10-141132 

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152  9-908781! 

9 

52 

9-767824 

291 

•232176 

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443 

10-140866 

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9-908690: 

8 

53 

9-767999 

291 

•232001 

9-859400 

443 

10-140600 

091401  152 

9-908599 

7 

54 

9-768173 

291 

•231827 

9-859666 

443 

10-140334 

091493:152 

9-908507 

6 

55 

9-768348 

291 

•231652 

9-859932 

443 

10-140068 

091584  152 

9-908416 

5 

56 

9-768522 

290 

•231478 

9-860198 

443 

10-139802 

091676 

153 

9-908324 

4 

57 

9-768697 

290 

•231303 

9-860464 

443 

10-139536 

091767 

153 

9-908233 

3 

58 

9-768871 

290 

•231129 

9-860730 

443 

10-139270 

091859 

153 

9-908141  2 

59 

9-769045 

290 

•230955 

9-860995 

443 

10-139005 

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9-908049!  1 

60 

9-769219 

290 

•230781 

9-861261 

443 

10-138739 

092042153 

9-907958:  0 

' 

Coalite. 

Secant. 

Cotangent. 

|   Tangent.   Cosecant.        Sine.   l|  ' 

54  DEG. 


LOGARITHMIC    SINES,  ETC. 


77 


36  DEO. 


• 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

?oJ'   Cosine. 

/ 

~~0~ 

9-769219 

•230781 

9-861261 

10-138739 

•092042 

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60 

1 

9-769393 

290 

•230607 

9-861527 

443 

10-138473 

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1539-907866 

59 

2 

9-769566 

289 

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9-861792 

443 

10-138208 

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1539-907774 

58 

3 

9-769740 

289 

•230260 

9-862058 

443 

10-137942 

•092318 

153  9-907682 

57 

4 

9-769913 

289 

•230087 

9-862323 

442 

10-137677 

•092410 

153:9-907590 

56 

5 

9-770087 

289 

•229913 

9-862589 

442 

10-137411 

•092502 

153  9-907498IJ55 

6 

9-770260 

289 

•229740 

9-862854 

442 

10-137146 

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1539-907406J54 

7 

9-770433 

288 

•229567 

9-863119 

442 

10-136881 

•092686 

153:9-907314!  53 

8 

9-770606 

288 

•229394 

9-863385 

442 

10-136615 

•092778 

154  9-907222 

52 

9 

9-770779 

288 

•229221 

9-863650 

442 

10-136350 

•092871 

154  9-9071291 

51 

10 

9-7709521  288 

•229048 

9-863915 

442 

10-136085 

•092963 

154:9-907037||50 

11 

9-771125 

288 

•228875 

9-864180 

442 

10-135820 

•093055 

154'9-906945!49 

12 

9-771298 

288 

•228702 

9-864445 

442 

10-135555 

•093148 

154  9-906852  i48 

13 

9-771470 

287 

•228530 

9-864710 

442 

10-135290 

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154  9-906760  147 

14 

9-771643 

287 

•228357 

9-864975 

442 

10-135025 

•093333 

1549-906667  46 

15 

9-771815 

287 

•228185 

9-865240 

442 

10-134760 

•093425 

154  9-906575;  45 

16 

9-771987 

287 

•228013 

9-865505 

441 

10-134495 

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154  9-906482!  44 

17 

9-772159 

287 

•227841 

9-865770 

441 

10-134230 

•093611 

154  9-906389  43 

18 

9-772331 

287 

•227669 

9-866035 

441 

10-133965 

•093704 

155|9-906296  42 

19 

9-772503 

286 

•227497 

9-866300 

441 

10-133700 

•093796 

1559-906204J|41 

20 

9-772675 

286 

•227325 

9-866564 

441 

10-133436 

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1559-906111140 

21 

9-772847 

286 

•227153 

9-866829 

441 

10-133171 

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155  9-90601  8;  39 

22 

9-773018 

286 

•226982 

9-867094 

441 

10-132906 

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1559-905925i38 

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9-773190 

286 

•226810 

9-867358 

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10-132642 

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15519-905832  37 

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286 

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10-132377 

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155  9-905739  36 

25 

9-773533 

285 

•226467 

9-867887 

441 

10-132113 

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155  9-905645 

35 

26 

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10-131848 

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34 

27 

9-773875 

285 

•226125 

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10-131584 

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155  9-905459 

33 

28 

9-774046 

285 

•225954 

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10-131320 

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29; 

9-774217 

285 

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10-131055 

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156  9-905272 

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30 

9-774388 

285 

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10-130791 

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156 

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10-130527 

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156 

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10-130263 

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156 

9-904992 

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10-129999 

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9-904898 

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10-129735 

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156 

9-904804  26 

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9-775240 

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10-129471 

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9-904711  125 

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10-129207 

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9-904617 

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9-904523 

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10-128151 

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10-127888 

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10-126570 

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10-126306 

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10-126043 

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9-903487 

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9-777613 

281 

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439 

10-125780 

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9-903392 

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9-777781 

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10-125516 

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9-903298 

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51 

9-777950 

281 

•222050 

9-874747 

439 

10-125253 

•096797 

158 

9-903203 

9 

52 

9-778119 

281 

•221881 

9-875010 

439 

10-124990 

•096892 

158 

9-903108 

8 

53 

9-778287 

281 

•221713 

9-875273 

439 

10-124727 

•096986 

158 

9-903014 

7 

54 

9-778455 

280 

•221545 

9-875536 

439 

10-124464 

•097081 

158i9-902919 

6 

55 

9-778624 

280 

•221376 

9-875800 

439 

10-124200 

•097176 

15819-902824 

5 

56 

9-778792 

280 

•221208 

9-876063 

438 

10-123937 

•097271 

158|9-902729 

4 

57 

9-778960 

280 

•221040 

9-876326 

438 

10-123674 

•097366 

15819-902634 

3 

58 

9-779128 

280 

•220872 

9-876589 

438 

10-123411 

•097461 

158!  9-902539 

2 

59 

9-779295 

280 

•220705 

9-876851 

438 

10-123149 

•097556 

159 

9-902444 

1 

60 

9-779463 

279 

•220537  9-877114 

438 

10-122886 

•097651 

159 

9-902349 

0 

'  ' 

Cosine. 

Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

53  DKU. 


78 


LOGARITHMIC    SINES,  ETC. 


37  DEG. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

0 

9-779463 

•220537 

9-877114 

10-122886 

•097651 

9^902349  60~ 

1 

9-779631 

279 

•220369 

9-877377  438 

10-122623 

•097747 

159 

9-902253  >59 

2 

9-779798 

279 

•220202 

9-877640  438 

10-122360 

•097842 

159 

9-902158:i58 

3 

9-779966 

279 

•220034 

9-877903 

438 

10-122097 

•Q97937 

159 

9-902063  57 

4 

9-780133 

279 

•219867 

9-878165 

438 

10-121835 

•098033 

159 

9-901967  50 

5 

9-780300 

279 

•219700 

9-878428  438  |10-121572 

•098128 

159 

9-901872  55 

6 

9-780467 

278 

•219533 

9-878691:  438 

10-121309 

•098224 

159 

9-901776154 

7 

9-780634 

278 

•219366 

9-878953  438 

10-121047 

•098319 

159 

9-901681  53 

8 

9-780801 

278 

•219199 

9-879216  437 

10-120784 

•098415 

159 

9-901585  52 

9 

9-780968 

278 

•219032 

9-879478  437 

10-120522 

•098510 

1599-901490  51 

10 

9-781134 

278 

•218866 

9-879741  i  437 

10-120259 

•098606 

159  9-901394  50 

11 

9-781301 

278 

•218699  9-880003  437 

10-119997 

•098702 

160 

9-901298  49 

12 

9-781468 

277 

•218532  ||  9-880265  437 

10-119735 

•098798 

160 

9-901202 

48 

13 

9-781634 

277 

•218366  9-880528!  437 

10-119472 

•098894 

160 

9-901106 

47 

14 

9-781800 

277 

•218200  19-880790  437 

10-119210 

•098990  160 

9-901010 

46 

15 

9-781966 

277 

•218034 

9-881052  437 

10-118948 

•099086  160 

9-900914 

45 

16 

9-782132 

277 

•217868 

9-881314  437  10-118686 

•099182 

160 

9-900818  44 

17 

9-782298 

277 

•217702 

9-881576 

437 

10-118424 

•099278  160 

9-900722  43 

18 

9-782464 

276 

•217536 

9-881839 

437 

10-118161 

•099374 

160 

9-900626  42 

19 

9-782630 

276 

•217370 

9-882101J  437 

10-117899 

•099471 

1609-900529  41 

20 

9-782796 

276 

•217204 

9-8823631  437 

10-117637 

•099567  160  9-900433  40 

21 

9-782961 

276 

•217039 

9-882625 

437 

10-117375 

•099663  161  9-900337  39 

22 

9-783127 

276 

•216873 

9-882887 

436110-117113 

•099760|161  9-900240  38 

23 

9-783292 

276 

•216708 

9-883148 

436 

10-116852 

•099856  161  J9-900144  37 

24 

9-783458 

275 

•216542 

9-883410 

436ilO-116590 

•099953  161  19-900047  36 

25 

9-783623 

275 

•216377 

9-883672 

436 

10-116328 

•100049!  161  9-899951  35 

26 

9-783788 

275 

•216212 

9-883934 

436 

10-116066 

•100146 

161 

9-899854  34 

27 

9-783953 

275 

•216047 

9-884196 

436 

10-115804 

•100243 

161 

9-899757  33 

28 

9-784118 

275 

•215882 

9-884457 

436 

10-115543 

•100340 

161 

9-899660  32 

29 

9-784282 

275 

•215718 

9-884719 

436 

10-115281 

•100436 

161 

9-899564  31 

30 

9-784447 

274 

•215553 

9-884980 

436 

10-115020 

•100533 

161 

9  -899467|  30 

31 

9-784612 

274 

•215388 

9-885242 

436 

10-114758  j-100630 

162 

9-899370  29 

32 

9-784776 

274 

•215224  9-885503 

436 

10-114497 

•100727 

162 

9-899273  28 

33 

9-784941 

274 

•215059 

9-885765 

436 

10-114235 

•100824 

162 

9-899176!i27 

34 

9-785105 

274 

•214895 

9-886026 

436 

10-113974 

•100922 

162 

9-899078  26 

35 

9-785269 

274 

•214731 

I  9-886288 

436 

10-113712 

•101019 

162 

9-898981  25 

36 

9-785433 

273 

•214567 

!  9-886549 

436 

10-113451 

•101116 

162 

9-898884124 

37 

9-785597 

273 

•214403  |  9-886810 

436 

10-113190 

•101213 

162 

9-898787 

23 

38 

9-785761 

273 

•214239J  9-887072 

435 

10-112928 

•101311 

162 

9-898689 

22 

39 

9-785925 

273 

•214075 

9-887333 

435 

10-112667 

•101408 

162 

9-898592 

21 

40 

9-786089 

273 

•213911 

9-887594 

435 

10-112406 

•101506 

162 

9-898494 

20 

41 

9-786252 

273 

•213748 

9-887855 

435 

10-112145 

•101603 

163 

9-898397 

19 

42 

9-786416 

272 

•213584 

9-888116 

435 

10-111884 

•101701 

163 

9-898299 

18 

43 

9-786579 

272 

•213421 

9-888377 

435 

10-111623 

•101798 

163 

9-898202 

17 

44 

9-786742 

272 

•213258 

9-888639 

435 

10-111361 

•101896 

163 

9-898104 

16 

45 

9-786906 

272 

•213094 

9-888900 

435 

10-111100 

•101994 

163 

9-898006 

15 

46 

9-787069 

272 

•212931 

9-889160 

435 

10-110840 

•102092 

163 

9-897908 

14 

47 

9-787232 

272 

•212768  9-889421 

435 

10-110579 

•102190 

163 

9-897810 

13 

48 

9-787395 

271 

•212605 

9-889682 

435 

10-110318 

•102288 

163 

9-897712 

12 

49 

9-787557 

271 

•212443 

9-889943 

435 

10-110057 

•102386 

163 

9-897614 

11 

50 

9-787720 

271 

•212280 

9-890204 

435 

10-109796 

•102484 

163 

9-897516 

10 

51 

9-787883 

271 

•212117 

9-890465 

435 

10-109535 

•102582 

163 

9-897418 

9 

52 

9-788045 

271 

•211955 

9-890725 

435 

10-109275 

•102680 

164 

9-897320 

8 

53 

9-788208 

271 

•211792 

9-890986 

434 

10-109014 

•102778 

164 

9-897222 

7 

54 
55 

9-788370 
9-788532 

271 
270 

•211630 
•211468 

9-891247 
9-891507 

434 
434 

10-108753 
10-108493 

•102877 
•102976 

164 
164 

9-897123 
9-897025 

6 
6 

56 

9-788694 

270 

•211306 

9-891768 

434 

10-108232 

•103074 

164 

9-896926 

4 

57 

9-788856 

270 

•211144 

9-892028 

434 

10-107972 

•103172 

164  9-896828 

3 

58 

9-789018 

270 

•210982 

9-892289 

434 

10-107711 

•103271 

164  9-896729 

2 

69 

9-789180 

270 

•210820 

9-892549 

434 

10-107451 

•103369 

164 

9-896631 

1 

60 

9-789342 

270 

•210658 

9-892810 

434 

10-107190 

•103468 

164 

9-896532 

0 

' 

Cosine. 

Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

52  DEG. 


LOGARITHMIC   SINES,  ETC. 


79 


DEG. 


' 

Sine. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

' 

0 

9-789342 

•210658 

9-892810 

10-107190 

•103468 

9-896532 

60 

1 

9-789504 

269 

•210496 

9-893070 

434 

10-106930 

•103567 

164 

9-896433 

59 

2 

9-789665 

269 

•210335 

9-893331 

434 

10-106669 

•103665 

165 

9-896335 

58 

3 

9-789827 

269 

•210173 

9-893591 

434 

10-106409 

•103764 

165 

9-896236 

57 

4 

9-789988 

269 

•210012 

9-893851 

434 

10-106149 

•103863 

165 

9-896137 

56 

5 

9-790149 

269 

•209851 

9-894111 

434 

10-105889 

•103962 

165 

9-896038 

55 

6 

9-790310 

269 

•209690 

9-894371 

434 

10-105629 

•104061 

165 

9-895939 

54 

7 

9-790471 

268 

•209529 

9-894632 

434 

10-105368 

•104160 

165 

9-895840 

53 

8 

9-790632 

268 

•209368 

9-894892 

434 

10-105108 

•104259 

165 

9-895741 

52 

9 

9-790793 

268 

•209207 

9-895152 

433 

10-104848 

•104359 

165 

9-895641 

51 

10 

9-790954 

268 

•209046 

9-895412 

433 

10-104588 

•104458 

165 

9-895542 

50 

11 

9-791115 

268 

•208885 

9-895672 

433 

10-104328 

•104557 

165 

9-8954431149 

12 

9-791275 

268 

•208725 

9-895932 

433 

10-104068 

•104657 

166 

9-895343 

48 

13 

9-791436 

267 

•208564 

9-896192 

433 

10-103808 

•104756 

166 

9-895244 

47 

14 

9-791596 

267 

•208404 

9-896452 

433 

10-103548 

•104855 

166 

9-895145 

46 

15 

9-791757 

267 

•208243 

9-896712 

433 

10-103288 

•104955 

166 

•9-895045 

45 

16 

9-791917 

267 

•208083 

9-896971 

433 

10-103029 

•105055 

166 

9-894945 

44 

17 

9-792077 

267 

•207923 

9-897231 

433  10-102769 

•105154 

166 

9-894846 

43 

18 

9-792237 

267 

•207763 

9-897491 

433110-102509 

•105254|166 

9-894746 

42 

19 

9-792397 

266 

•207603 

9-897751 

433  10-102249 

•105354il66 

9-894646 

41 

20 

9-792557 

266 

•207443 

9-898010 

433 

10-101990 

•105454 

166 

9-894546 

40 

21 

9-792716 

266 

•207284 

9-898270 

433 

10-101730 

•105554 

166 

9-894446'l39 

22 

9-792876 

266 

•207124 

9-898530 

433 

10-101470 

•105654 

167 

9-894346 

38 

23 

9-793035 

266 

•206965 

9-898789 

433 

10-101211 

•105754 

167 

9-894246 

37 

24 

9-793195 

266 

•206805 

9-899049 

433 

10-100951 

•105854J167 

9-894146 

36 

25 

9-793354 

265 

•206646 

9-899308 

432 

10-100692 

•105954 

167 

9-894046 

35 

26 

9-793514 

265 

•206486 

9-899568 

432 

10-100432 

•106054 

167 

9.-S93946 

34 

27 

9-793673 

265 

•206327 

9-899827 

432 

10-100173 

•106154 

167 

9-893846 

33 

28 

9-793832 

265 

•206168 

9-900086 

432 

10-099914 

•106255 

167 

9-893745 

32 

29 

9-793991 

265 

•206009 

9-900346 

432 

10-099654 

•106355 

167 

9-893645 

31 

30 

9-794150 

265 

•205850 

9-900605 

432 

10-099395 

•106456 

167 

9-893544 

30 

31 

9-794308 

264 

•205692 

9-900864 

432 

10-099136 

•106556 

167 

9-893444 

29 

32 

9-794467 

264 

•205533 

9-901124 

432 

10-098876 

•106657 

168 

9-893343 

28 

33 

9-794626 

264 

•205374 

9-901383 

432 

10-098617 

•106757 

168 

9-893243 

27 

34 

9-794784 

264 

•205216 

9-901642 

432 

10-098358 

•106858 

168 

9-893142 

26 

35 

9-794942 

264 

•205058 

9-901901 

432 

10-098099 

•106959 

168 

9-893041 

25 

36 

9-795101 

264 

•204899 

9-902160 

432 

10-097840 

•107060 

168 

9-892940 

24 

37 

9-795259 

264 

•204741 

9-902419 

432 

10-097581 

•107161 

168 

9-892839 

23 

38 

9-795417 

263 

•204583 

9-902679 

432 

10-097321 

•107261 

168 

9-892739 

22 

39 

9-795575 

263 

•204425 

9-902938 

432 

10-097062 

•107362 

168 

9-892638 

21 

40 

9-795733 

263 

•204267 

9-903197 

432 

10-096803 

•107464 

168 

9-892536 

20 

41 

9-795891 

263 

•204109 

9-903455 

432 

10-096545 

•107565 

168 

9-892435 

19 

42 

9-796049 

263 

•203951 

9-903714 

431 

10-096286 

•107666 

169 

9-892334 

18 

43 

9-796206 

263 

•203794 

9-903973 

431 

10-096027 

•107767 

169 

9-892283 

17 

44 

9-796364 

263 

•203636 

9-904232 

431 

10-095768 

•107868 

169 

9-892132 

16 

45 

9-796521 

262 

•203479 

9-904491 

431 

10-095509 

•107970 

169 

9-892030 

15 

46 

9-796679 

262 

•203321 

9-904750 

431 

10-095250 

•108071 

169 

9-891929 

14 

47 

9-796836 

262 

•203164 

9-905008 

431 

10-094992 

•108173 

169 

9-891827 

13 

48 

9-796993 

262 

•203007 

9-905267 

431 

10-094733 

•108274 

169 

9-891726 

12 

49 

9-797150 

262 

•202850 

9-905526 

431 

10-094474 

•108376 

169 

9-891624 

11 

50 

9-797307 

262 

•202693 

9-905784 

431 

10-094216 

•108477 

169 

9-891523 

10 

51 

9-797464 

261 

•202536 

i  9-906043 

431 

10-093957 

•108579 

169 

9-891421 

9 

52 

9-797621 

261 

•202379 

9-906302 

431 

10-093698 

•108681 

170 

9-891319 

8 

53 

9-797777 

261 

•202223 

9-906560 

431 

10-093440 

•108783 

170 

9-891217 

7 

54 

9-797934 

261 

•202066 

9-906819 

431 

10-093181 

•108885 

170 

9-891115 

6 

55 

9-798091 

261 

•201909 

9-907077 

431 

10-092923 

•108987 

170 

9-891013 

5 

56 

9-798247 

261 

•201753 

9-907336 

431 

10-092664 

•109089 

170 

9-890911 

4 

57 

9-798403 

261  -201597 

9-907594 

431 

10-092406 

•109191 

170 

9-890809 

3 

58 

9-798560 

260  -201440 

9-907852 

431 

10-092148 

•109293 

170 

9-890707 

2 

59 

9-798716 

260 

•201284 

!  9-908111 

431 

10-091889 

•109395 

170 

9-890605 

i  ! 

60 

9-798872 

260 

•201128 

9-908369 

431 

10-091631 

•109497 

170 

9-890503 

o  i 

—  ;  — 

ZJ 

Cosine. 

Secant,  i  Cotangent. 

Tangent. 

Cosecant. 

Sine. 

51  DBG. 


80 


LOGARITHMIC   SINES,  ETC. 


39  DEG. 


' 

Sine. 

5JJ)  |  Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff 
100 

Cosiiie. 

' 

0 

9-798872 

•201128 

9-908369 

10-091631 

•109497 

9-890503 

60 

1 

9-799028 

260 

•200972 

9-908628 

430 

10-091372 

•109600 

170 

9-890400 

59 

2 

9-799184 

260 

•200816 

9-908886 

430 

10-091114 

•109702 

171 

9-890298 

58 

3 

9-799339 

260 

•200661 

9-909144 

430 

10-090856 

•109805 

171 

9-890195 

57 

4 

9-799495 

259 

•200505 

9-909402 

430 

10-090598 

•109907 

171 

9-890093 

56 

6 

9-799651 

259 

•200349 

9-909660 

430 

10-090340 

•110010 

171 

9-889990 

55 

6 

9-799806 

2591  -200194 

9-909918 

430 

10-090082 

•110112 

171 

9-889888 

54 

7 

9-799962 

259 

•200038 

9-910177 

430 

10-089823 

•110215 

171 

9-889785 

53 

8 

9-800117 

259 

•199883 

9-910435 

430 

10-089565 

•110318 

171 

9-889682  152 

9 

9-800272 

259 

•199728 

9-910693 

430 

10-089307 

•110421 

171J9-889579  51 

10 

9-800427 

258 

•199573  !  9-910951 

430 

10-089049 

•110523 

171 

9-889477 

50 

11 

9-800582 

258 

•199418 

9-911209 

430 

10-088791 

•110626 

171 

9-889374 

49 

12 

9-800737 

258 

•199263 

9-911467 

430 

10-088533 

•110729 

172 

9-889271  148 

13 

9-800892 

258 

•199108 

9-911724 

430 

10-088276 

•110832 

172 

9-889168  47 

14 

9-801047 

258 

•198953 

9-911982 

430 

10-088018 

•110936 

172 

9-889064  46 

15 

9-801201 

258 

•198799 

9-912240 

430 

10-087760 

•111039 

172 

9-88896l!j45 

16 

9-801356 

258 

•198644 

9-912498 

430 

10-087502 

•111142 

172 

9-888858!  44 

17 

9-801511 

257 

•198489 

9-912756 

430 

10-087244 

•111245 

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9-872547113 

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9-823821 

235 

•176179 

9-951388 

424 

10-048612 

•127566 

188 

9-872434J  12 

49 

9-823963 

235 

•176037 

9-951642 

424 

10-048358 

•127679 

188 

9-872321  11 

50 

9-824104 

235 

•175896 

9-951896 

424 

10-048104 

•127792 

188 

9-872208:10 

51 

9-824245 

235 

•175755 

9-952150 

424 

10-047850 

•127905  188 

9-872095  9 

52 

9-824386 

235 

•175614 

9-952405 

424 

10-047595 

•128019S189 

9-871981  i  8 

53 

9-824527 

235 

•175473 

9-952659 

424 

10-047341 

•128132|l89 

9-871868 

i  7 

54 

9-824668 

235 

•175332 

9-952913 

424 

10-047087 

•128245 

189 

9-871755 

6 

55 

9-824808 

234 

•175192 

9-953167 

424 

10-046833 

•128359 

189 

9-871641 

5 

56 

9-824949 

234 

•175051 

9-953421 

423 

10-046579 

•128472 

189 

9-871528 

4- 

57 

9-825090 

234 

•174910 

9-953675 

423 

10-046325 

•128586:189 

9-871414 

3 

58 

9-825230 

234 

•174770 

9-953929 

423 

10-046071 

•128699189 

9-871301 

2 

59 

9-825371 

234 

•174629 

9-954183!  423 

10-045817 

•128813189 

9-871187 

1 

60 

9-825511 

234 

•174489 

9-954437!  423 

10-045563 

•128.927:189 

9-871073 

0 

' 

Cosine. 

Seuaut. 

;  Cotangent.  ' 

Tangent.   !  Cosecant. 

Sine. 

' 

48  DEG. 


LOGARITHMIC    SINES,  ETC. 


83 


42  DEC. 


' 

Siiie. 

Diff. 

100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 
100" 

Cosine. 

LT 

0 

9-825511 

•174489 

9-954437 

10-045563 

•128927 

9-871073 

1 

9-825651 

234 

•174349 

9-954691 

423 

10-045309 

•129040 

190 

9-870960  "59 

2 

9-825791 

233 

•174209 

9-954945 

423 

10-045055 

•129154 

190 

9-870846:'58 

3 

9-825931 

233 

•174069 

9-955200 

423 

10-044800 

•129268 

190 

9-870732  57 

4 

9-826071 

233 

•173929 

9-955454 

423 

10-044546 

•129382 

190 

9-870618  56 

5 

9-826211 

233 

•173789 

9-955707 

423 

10-044293 

•129496 

190 

9-870504  55 

6 

9-826351 

233 

•173649 

9-955961 

423 

10-044039 

•129610 

190 

9-870390  54 

7 

9-826491 

233 

•173509 

9-956215 

423 

10-043785 

•129724 

190 

9-870276  53 

8 

9-826631 

233 

•173369 

9-956469 

423 

10-043531 

•129839 

190 

9-870161  52 

9 

9-826770 

233 

•173230 

9-956723 

423 

10-043277 

•129953 

190 

9-870047  51 

10 

9-826910 

232 

•173090 

9-956977 

423 

10-043023 

•130067 

191 

9-869933  50 

11 

9-827049 

232 

•172951 

9-957231 

423 

10-042769 

•130182 

191 

9-869818  49 

12 

9-827189 

232 

•172811 

9-957485 

423 

10-042515 

•130296 

191 

9-869704  48 

13 

9-827328 

232 

•172672 

9-957739 

423 

10-042261 

•130411 

191 

9-869589  47 

14  9-827467 

232 

•172533 

9-957993 

423 

10-042007 

•130526 

191 

9-869474  46 

15  j  9-827606 

232 

•172394 

9-958246 

423 

10-041754 

•130640 

191 

9-869360  45 

16  9-827745 

232 

•172255 

9-958500 

423 

10-041500 

•130755 

191 

9-869245  44 

17  9-827884 

232 

•172116 

9-958754 

423 

10-041246 

•130870 

191 

9-869130  43 

18 

9-828023 

231 

•171977 

9-959008 

423 

10-040992 

•130985 

191 

9-869015  42  1 

19 

9-828162 

231 

•171838 

9-959262 

423 

10-040738 

•131100 

192 

9-868900  41 

20 

9-828301 

231 

•171699 

9-959516  423 

10-040484 

•131215 

192 

9-868785  40 

21 

9-828439 

231 

•171561 

9-959769 

423 

10.-040231 

•131330 

192 

9-868670  39 

22 

9-828578 

231 

•171422 

9-960023 

423 

10-039977 

•131445 

192 

9-868555  38 

23 

9-828716 

231 

•171284 

9-960277 

423 

10-039723 

•131560 

192 

9-868440,37 

24 

9-828855 

231 

•171145 

9-960531 

423 

10-039469 

•131676 

192 

9-868324;36 

25 

9-828993 

230 

•171007 

9-960784 

423 

10-039216 

•131791 

192 

9-868209  35 

26 

9-829131 

230 

•170869 

9-961038 

423 

10-038962 

•131907 

192 

9-868093:34 

27 

9-829269 

230 

•170731 

9-961291 

423 

10-038709 

•132022 

192 

9-867978  33 

28 

9-829407 

230 

•170593 

9-961545 

423 

10-038455 

•132138 

193 

9-867862:32 

29  1 

9-829545 

230 

•170455 

9-961799 

423 

10-038201 

•132253 

193 

9-867747'  31 

30  19-829(383 

230 

•170317 

9-962052 

423 

10-037948 

•132369 

193 

9-867631 

30 

31  9-829821 

230 

•170179 

9-962306 

423 

10-037694 

•132485 

193 

9-867515 

29 

32  1  9-829959 

229 

•170041 

9-962560 

423 

10-037440 

•132601 

193 

9-867399 

28 

33  ;  9-830097 

229 

•169903 

9-962813 

423 

10-037187 

•132717 

193 

9-867283 

27 

34  i  9-830234 

229 

•169766 

9-963067 

423 

10-036933 

•132833 

193 

9-867167 

26 

3o! 

9-830372 

229 

•169628 

9-963320 

423 

10-036680 

•132949 

193 

9-867051 

25 

36: 

9-830509 

229 

•169491 

9-963574 

423 

10-036426 

•133065 

193 

9-866935 

24 

37  9-830646 

229 

•169354 

9-963827 

423 

10-036173 

•133181 

194 

9-866819 

23 

38  9-830784 

229 

•169216 

9-964081 

423 

10-035919 

•133297 

194 

9-866703 

22 

39  9-830921 

229 

•169079 

9-964335 

423 

10-035665 

•133414 

194 

9-866586 

21 

40  19-831058 

228 

•168942 

9-964588 

423 

10-035412 

•133530 

194 

9-866470 

20 

41  !  9-831195 

228 

•168805 

9-964842 

423 

10-035158 

•133647 

194 

9-866353 

19 

42  19-831332 

228 

•168668 

9-965095 

422  110-034905 

•133763 

194 

9-866237 

18 

43  9-831469 

228 

•168531 

9-965349 

422 

10-034651 

•133880 

194 

9-866120 

17 

44 

9-831606 

228 

•168394 

9-965602 

422 

10-034398 

•133996 

194 

9-866004 

16 

45! 

9-831742 

228 

•168258 

9-965855 

422 

10-034145 

•134113 

195 

9-865887 

15 

46  9-831879 

228 

•168121 

9-966109 

422 

10-033891 

•134230 

195 

9-865770 

14 

47 

9-832015 

228 

•167985 

9-966362 

422 

10-033638 

•134347 

195 

9-865653 

13 

48 

9-832152 

227 

•167848 

9-966616 

422 

10-033384 

•134464 

195 

9-865536 

12 

49 

9-832288 

227 

•167712 

9-966869 

422 

10-033131 

•134581 

195 

9-865419 

11 

50 

9-832425 

227 

•167575 

9-967123 

422 

10-032877 

•134698 

195 

9-865302 

10 

51 

9-832561 

227 

•167439 

9-967376 

422 

10-032624 

•134815 

195 

9-865185 

9 

52 

9-832697 

227 

•167303 

9-967629 

422 

10-032371 

•134932 

195 

9-865068 

8 

53 

9-832833 

227 

•167167 

9-967883 

422 

10-032117 

•135050 

195 

9-864950 

7 

54 

9-832969 

227 

•167031 

9-968136 

422 

10-031864 

•135167 

195 

9-864833 

6 

55 

9-833105 

226 

•166895 

9-968389 

422 

10-031611 

•135284 

196 

9-864716 

5 

56 

9-833241 

226 

•166759 

9-968643 

422 

10-031357 

•135402 

19619-864598 

4 

57 

9-833377 

226 

•166623 

9-968896 

422 

10-031104 

•135519 

196 

9-864481 

3 

58 

9-833512 

226 

•166488 

9-969149 

422 

10-030851 

•135637 

196 

9-864363 

2 

59 

9-833648 

226 

•166352 

9-969403 

422 

10-030597 

•135755 

196 

9-864245 

1 

60 

9-833783 

226 

•166217 

9-969656 

422 

10-030344 

•135873 

196 

9-864127 

0 

'  ! 

Cosine. 

;   Secant.    Cotangent. 

Tangent.   Cosecant. 

Sine. 

' 

87 


47  DEG. 


84 


LOGARITHMIC    SINES,  ETC. 


' 

Sine. 

Diff. 
100" 

Cosecant. 

Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff. 

100" 

Cosine. 

' 

0 

9-833783 

•166217 

9-969656 

10-030344 

135873 

9-864127 

60 

1 

J-833919 

226 

•166081 

9-969909 

422 

10-030091 

135990 

196 

9-864010 

59 

2 

9-834054 

225 

•165946 

9-970162 

422 

10-029838 

136108 

196 

9-863892 

58 

3 

9-834189 

225 

•165811 

9-970416 

422 

10-029584 

136226 

197 

9-863774 

57 

4- 

9-834325 

225 

•165675 

9-970669 

422 

10-029331 

136344 

197 

9-863656 

56 

5 

9-834460 

225 

•165540 

9-970922 

422 

10-029078 

•136462 

197 

9-863538! 

55 

6 

9-834595 

225 

•165405 

9-971175 

422 

10-028825 

•136581 

197 

9-863419 

54 

7 

9-834730 

225 

•165270 

9-971429 

422 

10-028571 

•136699 

197 

9-863301 

53 

8 

9-834865 

225 

•165135 

9-971682 

422 

10-028318 

•136817 

197 

9-863183 

52 

9 

9-834999 

225 

•165001 

9-971935 

422 

10-028065 

•136936 

197 

9-863064 

51 

10 

9-835134 

224 

•164866 

9-972188 

422 

10-027812 

•137054 

197 

9-862946 

50 

11 

9-835269 

224 

•164731 

9-972441 

422 

10-027559 

•137173 

198 

9-862827 

49 

12 

9-835403 

224 

•164597 

9-972694 

422 

10-027306 

•137291 

198 

9-862709 

48 

13 

9-835538 

224 

•164462 

9-972948 

422 

10-027052 

•137410 

198 

9-862590 

47 

14 

9-835672 

224 

•164328 

9-973201 

422 

10-026799 

•137529 

198 

9-862471 

46 

15 

9-835807 

224 

•164193 

9-973454 

422 

10-026546 

•137647 

198 

9-862353 

45 

16 

9-835941 

224 

•164059 

9-973707 

422 

10-026293 

•137766 

198 

9-862234 

44 

17 

9-836075 

224 

•163925 

9-973960 

422 

10-026040 

•137885 

198 

9-862115 

43 

18 

9-836209 

223 

•163791 

9-974213 

422 

10-025787 

•138004 

198 

9-861996 

42 

19 

9-836343 

223 

•163657 

9-974466 

422 

10-025534 

•138123 

198 

9-861877 

41 

20 

9-836477 

223 

•163523 

9-974719 

422 

10-025281 

•138242 

198 

9-861758 

40 

21 

9-836611 

223 

•163389 

9-974973 

422 

10-025027 

•138362 

199 

9-861638 

39 

22 

9-836745 

223 

•163255 

9-975226 

422 

10-024774 

•138481 

199 

9-861519 

38 

23 

9-836878 

223 

•163122 

9-975479 

422 

10-024521 

•138600 

199 

9-861400 

37 

24 

9-837012 

223 

•162988 

9-975732 

422 

10-024268 

•138720 

199 

9-861280 

36 

25 

9-837146 

222 

•162854 

9-975985 

422 

10-024015 

•138839 

199 

9-861161 

35 

26 

9-837279 

222 

•162721 

9-976238 

422 

10-023762 

•138959 

199 

9-861041134 

27 

9-837412 

222 

•162588 

9-976491 

422 

10-023509 

•139078 

199 

9-860922  33 

28 

9-837546 

222 

•162454 

9-976744 

422 

10-023256 

•139198 

199 

9-860802  32 

29 

9-837679 

222 

•162321 

9-976997 

422 

10-023003 

•139318 

199 

9-860682]  31 

30 

9-837812 

222 

•162188 

9-977250 

422 

10-022750 

•139438 

200 

9-860562  '30 

31 

9-837945 

222 

•162055 

9-977503 

422 

10-022497 

•139558 

200 

9-860442  29 

32 

9-838078 

222 

•161922 

9-977756 

422 

10-022244 

•139678 

200 

9-860322  28 

33 

9-838211 

221 

•161789 

9-978009 

422 

10-021991 

•139798 

200 

9-860202  27 

34 

9-838344 

221 

•161656 

9-978262 

422 

10-021738 

•139918 

200 

9-860082'  26 

35 

9-838477 

221 

•161523 

9-978515 

422 

10-021485 

•140038 

200 

9-859962 

25 

36 

9-838610 

221 

•161390 

9-978768 

422 

10-021232 

•140158 

200 

9-859842 

24 

37 

9-838742 

221 

•161258 

9-979021 

422 

10-020979 

•140279 

200 

9-859721 

23 

38 

9-838875 

221 

•161125 

9-979274 

422 

10-020726 

•140399 

201 

9-859601 

22 

39 

9-839007 

221 

•160993 

9-979527 

A22 

10-020473 

•140520 

201 

9-859480 

21 

40 

9-839140 

221 

•160860 

9-979780 

422 

10-020220 

•140640 

201 

9-859360 

20 

41 

9-839272 

220 

•160728 

9-980033 

422 

10-019967 

•140761 

201 

9-859239 

19 

42 

9-839404 

220 

•160596 

9-980286 

422 

10-019714 

•140881 

201 

9-859119 

18 

43 

9-839536 

220 

•160464 

9-980538 

422 

10-019462 

•141002 

201 

9-858998 

17 

44 

9-839668 

220 

•160332 

9-980791 

422 

10-019209 

•141123 

201 

9-858877 

16 

45 

9-839800 

220 

•160200 

9-981044 

422 

10-018956 

•141244 

201 

9-858756 

15 

46 

9-839932 

220 

•160068 

9-981297 

422 

10-018703 

•141365 

202 

9-858635 

14 

47 

9-840064 

220 

•159936 

9-981550 

422 

10-018450 

•141486 

202 

9-858514 

13 

48 

9-840196 

219 

•159804 

9-981803 

422 

10-018197 

•141607 

202 

9-858393 

12 

49 

9-840328 

219 

•159672 

9-982056 

422 

10-017944 

•141728 

202 

9-858272 

11 

50 

9'840459 

219 

•159541 

9-982309 

422 

10-017691 

•141849 

202 

9-858151 

10 

51 

9-840591 

219 

•159409 

9-982562 

421 

10-017438 

•141971 

202 

9-858029 

9 

52 

9-840722 

219 

•159278 

9-982814 

421 

10-017186 

•142092 

202 

9-857908 

8 

53 

9-840854 

219 

•159146 

9-983067 

421 

10-016933 

•142214 

202 

9-857786 

7 

54 

9-840985 

219 

•159015 

9-983320 

421 

10-016680 

•142335 

202 

9-857665 

6 

55 

9-841116 

219 

•158884 

9-983573 

421 

10-016427 

•142457 

203 

9-857543 

5 

56 

9-841247 

218 

•158753 

9-983826 

421 

10-016174 

•142578 

203 

9-857422 

4 

57 

9-841378 

218 

•158622 

9-984079 

421 

10-015921 

•142700 

203 

9-857300 

3 

58 

9-841509 

218 

•158491 

9-984331 

421  |10-015669 

•142822 

203 

9-857178 

2 

59 

9-841640 

218 

•158360 

9-984584 

421 

10-015416 

•142944 

203 

9-857056 

1 

60 

9-841771 

218 

•158229 

9-984837 

421 

10-015163 

•143066 

203 

9-856934 

0 

' 

Cosine. 

Secant.    Cotangent. 

Tangent. 

Cosecant. 

Sine. 

' 

DEC. 


LOGARITHMIC    SINES,  ETC. 


85 


44  DEC. 


Sine. 

^y>  Cosecant.    Tangent. 

Diff. 
100" 

Cotangent. 

Secant. 

Diff 

100'' 

Cosine. 

F 

0 

9-841771 

i  -158229  jj  9-984837 

I0-Of5163 

•143066 

9-856934 

60 

1 

9-841902 

218  -1580981  9-985090 

421 

10-014910 

•143188 

203 

9-856812 

!59 

2 

9-842033 

218  ;  -157967  9-985343 

421 

10-014657 

•143310 

203 

9-856690 

|58 

3 

9-842163 

218  -157837  [j  9-985596 

421 

10-014404 

•143432  204  9-856568|!57 

4 

9-842294 

217 

•157706  >  9-985848 

421 

10-014152 

•143554 

204 

9-866446]66 

5 

9-842424 

217 

•157576"  9-986101 

421 

10-013899 

•143677 

204 

9-856323 

55 

6 

9-842555 

217 

•157445  l\  9-986354 

421 

10-013646 

•143799 

204 

9-856201 

54 

7 

9-842685 

217 

•157315"  9-986607 

421 

10-013393 

•143922 

204(9-856078 

53 

8 

9-842815 

217 

•157185;  9-986860 

421 

10-013140 

•144044 

204  9-855956 

52 

9 

9-842946 

217 

•157054  9-987112 

421 

10-012888 

•144167 

204(9-855833 

51 

10 

9-843076 

217 

•156924  '!  9-987365 

421 

10-012635 

•144289 

204  9-855711 

50 

11 

9-843206 

217 

•156794  9-987618 

421 

10-012382 

•144412 

205  9-855588 

49 

12 

9-843336 

216 

•156664  9-987871 

421 

10-012129 

•144535 

205  9-855465 

48 

13 

9-843466 

216 

•156534  9-988123 

421 

10-011877 

•144658  205 

9-855342 

47 

14 

9-843595 

216 

•156405  9-988376 

421 

10-011624 

•144781  1205 

9-855212 

46 

15 

9-843725 

216 

•156275  9-988629 

421 

10-011371 

•144904 

205 

9-855096 

45 

16 

9-843855 

216 

•156145i  9-988882 

421 

10-011118 

•145027 

205 

9-854973 

44 

17 

9-843984 

216 

•156016 

9-9891341  421 

10-010866 

•145150 

205 

9-854850 

43 

18 

9-844114 

216 

•155886 

9-989387 

421 

10-010613 

•145273 

205 

9-854727 

42 

19 

9-844243 

216 

•155757 

9-989640 

421 

10-010360 

•145397 

206 

9-854603 

41 

20 

9-844372 

215 

•155628  9-989893 

421 

10-010107 

•145520 

206 

9-854480 

40 

21 

9-844502 

215 

•155498  9-990145 

421 

10-009855 

•145644 

206 

9-854356 

39 

22 

9-844631 

215 

•155369  9-990398 

421 

10-009602 

•145767 

206 

9-854233  |38 

23 

9-844760 

215 

•155240  9-990651 

421 

10-009349 

•145891 

206 

9-854109 

37 

24 

9-844889 

215 

•155111  9-990903 

421 

10-009097 

•146014 

206  9-853986 

be 

25  l|  9-845018 

215 

•154982  9-991156 

421 

10-008844 

•146138 

206 

9-853862 

J35 

26 

9-845147 

215 

•154853  9-991409 

421 

10-008591 

•146262 

206 

9-853738 

134 

27 

9-845276 

215 

•154724!  9-991662 

421 

10-008338 

•146386  206 

9-853614 

33 

28 

9-845405 

214 

•154595  9-991914  421 

10-008086 

•146510 

207 

9-853490 

32 

29 

9-845533 

214 

•154467  9-992167  421 

10-007833 

•146634 

207 

9-853366!|31 

30 
31 

9-845662 
9-845790 

214 

214 

•154338  9-992420 
•154210  9-992672 

421 
421 

10-007580 
10-007328 

•146758207 
•1468821207 

9-853242  30 
9-853118  29 

32 

9-845919 

214 

•154081  j|  9-992925  421 

10-007075 

•147006 

207 

9-852994 

28 

33  II  9-846047!  214 

•153953 

9-993178  421 

10-006822 

•147131 

207 

9-852869 

27 

34 

9-846175  214 

•153825 

9-993430  421 

10-006570 

•147255 

207 

9-852745 

26 

35 

i  9-846304  214 

•153696 

9-993683 

421 

10-006317 

•147380 

207 

9-852620 

25 

36 

1  9-846432  214 

•153568 

9-993936 

421 

10-006064 

•147504 

207 

9-852496 

24 

37 

9-846560  213 

•153440 

9-994189 

421 

10-005811 

•147629 

208 

9-852371 

23 

38 

9-8466881  213 

•153312 

9-994441 

421 

10-005559 

•147753 

208 

9-852247 

22 

39 

9-846816  213 

•153184 

9-994694 

421 

10-005306 

•147878 

208 

9-852122 

21 

40 

i  9-846944  213 

•153056 

9-994947 

421 

10-005053 

•148003 

208 

9-851997 

20 

41 

9-847071  213 

•152929 

9-995199 

421 

10-004801 

•148128 

208 

9-85,1872 

19 

42 

9-847199  213 

•152801 

9-995452 

421 

10-004548 

•148253 

208 

9-851747 

18 

43 

9-847327i  213 

•152673 

9-995705  421 

10-004295 

•148378 

208 

9-851622  17 

44 

9-847454  213 

•152546 

9-995957  421 

10-004043 

•148503 

208 

9-861497  16 

45 

9-847582'  212 

•152418 

9-996210  421 

10-003790 

•148628 

209 

9-851372!  15 

46 

9-847709  2l2 

•152291 

9-996463 

421 

10-003537 

•148754 

2099-851246 

14 

47 

9-847836  212 

•152164 

9-996715 

421 

10-003285 

•148879 

209  9-851121 

13 

48 

9-847964  212 

•152036 

9-996968 

421 

10-003032 

•149004 

20919-850996 

12 

49 

9-848091  i  212 

•151909 

9-997221 

421 

10-002779 

•149130 

209J9-850870 

11 

50 

9-848218  212 

•151782 

9-997473 

421 

10-002527 

•149255 

209  9-850745 

10 

51 

9-848345!  212 

•151655 

9-997726 

421 

10-002274 

•149381 

209!9-850619 

9 

52 

9  -848472  !  212 

•151528 

9-997979 

421 

10-002021 

•149507 

2099-850493 

8 

53 

9-848599!  211 

•151401 

9-998231 

421 

10-001769 

•149632 

2109-850368 

7 

54 

9-848726  211 

•151274 

9-998484 

421 

10-001516 

•149758 

210 

9-850242 

6 

55  9-848852  211 

•151148 

9-998737 

421 

10-001263 

•149884J210 

9-850116 

5 

56  !  9-848979  211 

•151021 

9-998989 

421 

10-001011  1-150010  210 

9-849990 

4s 

57 

9-849106  211 

•150894 

9-999242 

421 

10-000758  11-150136  210  9-849864 

3 

58 

9-849232  211 

•150768 

9-999495 

421 

10-000505 

•150262!210l9-849738 

2 

59  9-849359  211 

•150641  1  9-999747 

421 

10-000253 

•150389210:9-849611  1  1 

GO  9-849485  211 

•150515  10-000000 

421 

10-000000 

•1505152109-849485 

0 

'   '   Cosine. 

Secant.    Cotangent. 

Tangent.   !  Cosecant.        Sine. 

' 

45  DKG. 


INDEX. 


ABBREVIATION  of  the  reduction  of  decimals,  17. 

Abrasion,  limits  of,  301. 

Absolute  resistances,  288. 

Absolute  strength  of  cylindrical  columns,  274. 

Accelerated  motion,  386. 

Accelerated  motion  of  wheel  and  axle,  419. 

Acceleration,  415. 

Acceleration  and  mass,  422. 

Actual  and  nominal  horse  power,  240. 

Addition  of  decimals,  22. 

Addition  of  fractions,  20. 

Adhesion,  297. 

Air,  expansion  of,  by  heat,  173. 

Air  that  passes  through  the  fire  for  each  horse 
power  of  the  engine,  210. 

Air,  water,  and  mercury,  355. 

Air-pump,  254. 

Air-pump,  diameter  of,  eye  of  air-pump  cross 
head,  145. 

Air-pump  machinery,  dimensions  of  several 
parts  of,  144. 

Air-pump  strap  at  and  below  cutter,  147. 

Air-pump  studs,  144. 

Ale  and  beer  measure,  8. 

Algebra  and  arithmetic,  characters  used  in,  12. 

Algebraic  quantities,  134. 

Alloys,  strength  of,  287. 

Ambiguous  cases  in  spherical  trigonometry, 
381. 

Amount  of  effective  power  produced  by  steam, 
266. 

Anchor  rings,  90. 

Angle  iron,  91,  408,  409,  410. 

Angles  of  windmill  sails,  445. 

Angles,  measurement  of,  by  compasses  only, 
382. 

Angular  magnitudes,  359. 

Angular  magnitudes,  how  measured,  373, 

Angular  velocity,  412. 

Apothecaries'  weight,  6. 

Apparent  motion  of  the  stars,  353. 

Application  of  logarithms,  334. 

Approximating  rule  to  find  the  area  of  a  seg- 
ment of  a  circle,  67. 

Approximations  for  facilitating  calculations, 
55. 

Arc  of  a  circle,  to  find,  49. 

Arc  of  one  minute,  to  find  the  length  of,  361. 

Arc,  the  length  of  which  is  equal  to  the  ra- 
dius, 357. 

Architecture,  naval,  453. 

Arcs,  circular,  to  find  the  lengths  of,  68. 

Area  of  segment  and  sector  of  a  circle,  51. 

Area  of  steam  passages,  220. 

Areas  of  circles,  57. 

Areas  of  segments  and  zones  of  circles,  64, 
65,  66,  67. 

Arithmetic,  10. 

Arithmetical  progression,  to  find  the  square 
root  of  numbers  in,  126. 

Arithmetical  solution  of  plane  triangles,  366. 


Arithmetical  proportion  and  progression,  35 

to  38. 
Ascent  of  smoke  and  heated  air  in  chimneys, 

208. 
Atmospheres,  elastic  force  of  steam  in,  195, 

196. 

Atmospheric  air,  weight  of,  356. 
Average  specific  gravity  of  timber,  396. 
Avoirdupois  weight,  6. 
Axle  and  wheel,  417. 
Axle  of  locomotive  engine,  168,  169. 
Axle-ends  or  gudgeons,  301. 
Axles,  friction  of,  298,  300. 

BALLS  of  cast  iron,  407. 

Bands,  ropes,  <fcc.,  267. 

Bar  iron,  400. 

Beam,  151. 

Beam,  the  strongest,  276. 

Bearings  of  water  wheels,  285. 

Bearings  or  journals  for  shafts  of  various 
diameters,  287. 

Beaters  of  threshing  machine,  445. 

Before  and  behind  the  piston,  232. 

Blast  pipe,  171. 

Blistered  steel,  281. 

Blocks,  cords,  ropes,  shelves,  428. 

Bodies,  cohesive  power  of,  175. 

Bodies  moving  in  fluids,  324. 

Boiler,  171. 

Boiler  plate,  experiments  on,  at  high  tempe- 
ratures, 220. 

Boiler  plates,  403. 

Boilers,  256  and  257. 

Boilers  of  copper  and  iron,  diminution  of 
the  strength  of,  219. 

Boilers,  properties  of,  215. 

Boilers,  strength  of,  218. 

Bolts  and  nuts,  406. 

Bolts,  screw  and  rivet,  220. 

Boring  iron,  445. 

Bossut  and  Michelloti,  experiments  on  the 
discharge  of  water,  319. 

Boyle  of  Cork,  200. 

Bramah's  press,  427. 

Branch  steam-pipe,  148. 

Brass,  copper,  iron,  properties  of,  280. 

Brass,  round  and  square,  408. 

Breast  wheels,  328. 

Breast  and  overshot  wheels,  maximum  ve- 
locity of,  443. 

Buckets  and  shrouding  of  water  wheels,  446. 

Building,  to  support  with  cast  iron  columns, 
293. 

Bushel,  5. 

Butt  for  air-pump,  146. 

Butt,  thickness  and  breadth  of,  143. 

Butt,  to  find  the  breadth  of,  141. 

Byrne's  logarithmic  discovery,  340. 

Byrne's  theory  of  the  strength  of  materials, 
272. 

583 


584 


INDEX. 


CALCULATION  in  the  art  of  ship -building,  453 
to  494. 

Calculation  of  Friction,  267. 

Carriages,  motion  of,  on  inclined  planes,  429. 

Carriages  travelling  on  ordinary  roads,  307. 

Carrier  or  intermediate  wheels,  434. 

Carts  on  ordinary  roads,  311. 

Cases  in  plane  trigonometry,  363. 

Cast  iron,  174. 

Cast  iron  pipes,  404. 

Centre  of  effort,  483. 

Centre  of-  gravity,  175. 

Centre  of  gravity  of  displacement  of  a  ship, 
456,  457,  458. 

Centre  of  gyration,  180. 

Centre  of  oscillation,  187. 

Centres  of  bodies,  386. 

Centres  of  gravity,  gyration,  percussion  os- 
cillation, 391. 

Centripetal  and  centrifugal  forces,  178,  450. 

Chain  bridge,  412. 

Chimney,  171,  208,  257. 

Chimney,  size  of,  212. 

Chimney,  to  what  height  it  may  be  carried 
with  safety,  212. 

Circle,  calculations  respecting,  48,  49,  50,  53. 

Circle  of  gyration  in  water  wheels,  444. 

Circles,  57  to  81. 

Circles,  areas  of,  57  to  63. 

Circular  arcs,  68. 

Circular  motion,  422. 

Circular  parts  of  spherica?  triangles,  375. 

Circumference  of  a  circle  to  radius  1,  361. 

Circumferences  of  circles,  57. 

Cloth  measure,  7. 

Coefficient  of  efflux,  314. 

Coefficients  of  friction,  299. 

Cohesive  strength  of  bodies,  how  to  find,  281. 

Collision  of  railway  trains,  452. 

Columns,  comparative  strength  of,  294. 

Combinations  of  algebraic  quantities,  134. 

Common  fractions,  15. 

Common  materials,  280. 

Complementary  and  supplementary  arcs,  374. 

Compound  proportion,  14. 

Condenser,  226. 

Condensing  water,  223. 

Conduit  pipes,  discharge  by,  322. 

Cone,  82. 

Conical  pendulum,  185  to  187. 

Connecting  rod,  140,  141,  253. 

Continuous  circular  motion,  432. 

Contraction  by  efflux,  316. 

Contraction  of  the  fluid  vein,  313. 

Contractions  in  the  calculation  of  loga- 
rithms, 348. 

Copper  boilers,  219. 

Copper,  iron,  and  lead,  405. 

Cosine,  to  find,  361. 

Cosines,  contangents,  <fcc.,  for  every  degree 
and  minute  in  the  quadrant,  540  to  576. 

Cosines,  natural,  411. 

Cover  on  the  exhausting  side  of  the  valve, 
in  parts  of  the  length  of  stroke,  231. 

Cover  on  the  steam  side,  226. 

Crane,  427. 

Crane,  sustaining  weight  of,  285. 

Crank  at  paddle  centre,  135. 

Crank  axle,  diameter  of  the  outside  bearings 
of,  168. 


Crank  axle  of  locomotive,  169. 

Crank  pin,  170,  252. 

Crank  pin  journal,  252. 

Crank  pin  journal,  to  find  the  diameter  of,  139 

Crank  pin  journal,  to  find  the  length  of,  1 39. 

Cross  head,  252. 

Cross  head,  to  find  the  breadth  of  eye  of,  139 

Cross  head,  to  find  the  depth  of  eye  of,  139 

Cross  multiplication,  27. 

Cross  tail,  253. 

Cube,  79. 

Cube  and  cube  roots  of  numbers,  100  to  116. 

Cube  root  of  numbers  containing  decimals, 
128. 

Cube  root,  to  extract,  32. 

Cubes,  397  to  400. 

Cubes,  to  extend  the  table  of,  128. 

Curve,  to  find  the  length  of,  by  construction,  72. 

Curves,  to  find  the  areas  of,  453. 

Cuttings  and  embankments,  97. 

Cylinder  side  rods  at  ends,  to  find  the  diame- 
ter of,  143. 

Cylinders,  80,  397  to  400. 

Cylinders  of  cast  iron,  404. 

DAMS  inclined  to  the  horizon,  316. 

Decimal  approximations  for  facilitating  cal- 
culations, 55. 

Decimal  equivalents,  56. 

Decimal  fractions,  22. 

Decimal  fractions,  table  of,  73. 

Decimals,  addition  of,  22. 

Decimals,  division  of,  24. 

Decimals,  multiplication  of,  23. 

Decimals,  reduction  of,  25,  26. 

Decimals,  rule  of  three  in,  27. 

Decimals,  subtraction  of,  23. 

Deflection  of  beams,  295. 

Deflection  of  rectangular  beams,  294. 

Depth  of  web  at  the  centre  of  main  beam,  150. 

Destructive  effects  produced  by  carriages  on 
roads,  311. 

Devlin's  oil,  297. 

Diagram  of  a  curve  of  sectional  areas,  460. 

Diagram  of  indicator,  265. 

Diameter  of  cylinder,  251. 

Diameter  of  main  centre  journal,  143. 

Diameter  of  plain  part  of  crank  axle,  169. 

Diameter  of  the  outside  bearings  of  the  crank 
axle,  168. 

Diameters  of  wheels  at  their  pitch  circle  to 
contain  a  required  number  of  teeth,  436. 

Dimensions  of  the  several  parts  of  furnaces 
and  boilers,  254. 

Direct  method  to  calculate  the  logarithm  of 
any  number,  346. 

Direct  strain,  278. 

Discharge  by  compound  tubes,  321. 

Discharge  by  different  apertures  from  differ- 
ent heads  of  water,  318. 

Discharge  of  water,  446. 

Discharges  from  orifices,  426. 

Displacement  of  a  ship  when  treated  as  a 
floating  body,  455. 

Displacement  of  ships,  by  vertical  and  hori- 
zontal sections,  460,  494. 

Distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  exhausting  port  is  shut 
and  when  it  is  open,  231. 

Distances,  how  to  measure,  369. 


INDEX. 


585 


Division  by  logarithms,  336. 

Dodecaeclron,  89. 

Double  acting  engines,  rods  of,  250. 

Double  position,  44. 

Double  table  of  ordinates,  457. 

Drainage  of  water  through  pipes,  325. 

Dr.  Dalton,  and  his  countryman,  Dr.  Young, 

of  Dublin, 
Drums,  422. 

Drums  in  continuous  circular  motion,  432. 
Dry  or  corn  measure,  8. 
Duodecimals,  27. 
Dutch  sails  of  windmills,  333. 
D.  valves,  233. 
Dynamometer,  used  to  measure  force,  269. 

EDUCTION  ports,  171. 

Effective  discharge  of  water,  314. 

Effective  heating  surface  of  flue  boilers,  256. 

Effects  of  carriages  on  ordinary  roads,  311. 

Elastic  force  of  steam,  188. 

Elastic  fluids,  205. 

Elliptic  arcs,  69,  70,  71,  72. 

Embankments  and  cuttings,  97. 

Endless  screw,  431. 

Engineering  and  mechanical  materials,  386. 

Engine,  motion  of  steam  in,  206. 

Engine  tender  tank,  92. 

Enlargements  of  pipes,  interruption  of  dis- 
charge by,  321. 

Evolution,  29. 

Evolution  by  logarithms,  339. 

Eye,  diameter  of,  251. 

Eye  of  crank,  136. 

Eye  of  crank,  to  find  the  length  and  breadth 
of  large  and  small,  142. 

Eye  of  round  end  of  studs  of  lever,  143. 

Examples  on  the  velocity  of  wheels,  drums, 
and  pulleys,  438. 

Exhaust  port,  230. 

Expanded  steam,  236. 

Expansion,  237. 

Expansion,  economical  effect  of,  216. 

Experiments  on  the  strength  and  other  pro- 
perties of  cast  iron,  174. 

Explanation  of  characters,  12. 

Extended  theory  of  angular  magnitude,  374. 

Exterior  diameter  of  large  eye,  252. 

Extraction  of  roots  by  logarithms,  339. 

FALL  of  water,  444. 

Feed  pipe,  150. 

Feed  water,  222. 

Felloes  "of  wheels,  309. 

Fellowship,  or  partnership,  41. 

Fik-grate,  171,  214. 

Fitz|erald,  264,  269. 

Flange,  91. 

Flat  bar  iron,  407. 

Flat  iron,  400. 

Flexure  by  vertical  pressure,  292. 

Flexure  of  revolving  shafts,  pillars,  <fcc.,  296. 

Flues,  256. 

Flues,  fires,  and  boilers,  217. 

Fluids,  thr-motion  of  elastic,  205. 

Fluids,  to  find  the  specific  gravity  of,  392. 

Fluids,  the  pressure  of,  448. 

Fluid  vein,  contraction  of,  313. 

Foot-valve  passage,  149. 

Force,  267. 

Force,  loss  of,  in  steam  pipes,  221. 


Force  of  steam,  188. 

Forces,  centrifugal  and  centripetal,  178,  450. 

Fore  and  after  boTfies  of  immersion,  456,  460. 

Form,  the  strongest,  275. 

Formulas  for  the  strength  of  various  parts 
of  marine  engines,  251. 

Formulas  to  find  the  three  angles  of  a  sphe- 
rical triangle  when  the  three  sides  are 
given,  385. 

Formula,  very  useful,  271. 

Fourth  and  fifth  power  of  numbers,  129. 

Fractions,  common,  15. 

Fractions,  reduction  of,  16,  17,  IS,  19. 

Fractions,  addition  of,  20. 

Fractions,  subtraction  of,  21. 

Fractions,  multiplication  of,  21. 

Fractions,  division  of,  21. 

Fractions,  the  rule  of  three  in,  21. 

Fractions,  decimal,  22. 

Fractions,  table  of,  73. 

Fractions,  addition  contracted,  78. 

Fracture,  292. 

Franklin  Institute,  172,  219. 

French  litre,  355. 

French  measures,  5,  6. 

French  metre,  347. 

Friction,  238. 

Friction,  coefficents  of,  300. 

Friction  of  fluids,  325. 

Friction  of  rest  and  of  motion,  267. 

Friction  of  steam  engines  of  different  modi- 
fications, 302. 

Friction  of  water  against  the  sides  of  pipes,321. 

Friction  of  water-wheels,  windmills,  <fcc.,  267. 

Friction,  or  resistance  to  motion,  in  bodies 
rolling  or  rubbing  on  each  other,  297. 

Friction,  laws  of,  298. 

Frustums,  83. 

Frustum  of  spheroid,  87. 

Furnace,  256. 

Furnace  room,  213. 

GALLON,  5. 
Gases,  394. 
Geering,  422. 

General  and  universal  expression,  376. 
General  observations  on  the  steam  engine,259. 
General  trigonometrical  solutions,  365,  369. 
Geometrical  construction,  #62. 
Geometrical  construction  of  the  proportion 

of  the  radius  of  a  wheel  to  its  pitch,  440. 
Geometrical  proportion  and  progression,  38. 
Gibs  and  cutter,  140,253. 
Gibs  and  cutter  through  air  pump  cross-head, 

146,  147. 
Gibs    and    cutter    through    cross-tail    and 

through  butt,  141. 
Gibs  and  cutter,  to  find  the  thickness  and 

breadth  of,  143. 
Girder,  275. 

wGirth,  the  mean  in  measuring,  94. 
Glenie,  the  mathematician,  287. 
Globe,  85. 
Grate  surface,  255. 
Gravity,  centre  of,  175,  386. 
Gravity,  specific,  391. 
Gravity,  weight,  mass,  386. 
Gudgeons,  420. 
Gyration,  centre  of,  180,  390. 
Gyration,  the  centre  of  different  figures  and 

bodies,  181. 


586 


INDEX. 


HEADS  of  water,  318. 

Heating  surface,  256. 

Heating  surface  of  boiler?,  215. 

Heights  and  discharges  of  water,  319. 

Heights  and  distances,  359. 

Height  of  chimneys,  210. 

Height  of  metacentre,  489,  483. 

Hewn  and  sawed  timber,  95. 

Hexagon,  heptagon,  48. 

High  pressure  and  condensing  engines,  234. 

Hollow  shafts,  to  find  the  strength  of,  284. 

Horizontal  distance  of  centre  of  radius  bar, 

246,  247. 

Horse  power,  240. 
Horse  power  of  an  engine,  dimensions  made 

to  depend  upon  the  nominal  horse  power 

of  an  engine,  147. 

Horse  power  of  pumping  engines,  447. 
Horse  power,  tables  of,  243,  244. 
Hot  blast,  174. 
Hot  liquor  pumps,  446. 
Hydraulic  pressure  working  machinery,  330. 
Hydraulics,  267,  312. 
Hydrogen,  weight  of,  356. 
Hydrostatic  press,  448. 
Hyperboloid,  88. 

Hyperbolic  logarithms,  130  to  133. 
Hyperbolic  logarithms,  how  to  calculate,  353. 
Hypothenuse  of  a  spherical  triangle,  to  find, 

378. 
Hypothenuse,  47. 

ICOSAEDRON,  89. 

Immersed  portions  of  a  ship,  to  calculate, 
456. 

Immersion  and  emersion,  453  to  467. 

Impact,  449. 

Impinging  of  elastic  and  inelastic  bodies, 
452. 

Inaccessible  distances,  372. 

Inches  in  a  solid  foot,  96. 

Inclined  plane,  428,  429,  430. 

Inclination  of  the  traces  of  ordinary  car- 
riages, 311. 

Inclinations,  discharge  of 'a  6-inch  pipe  at 
several,  326. 

Increase  of  efficiency  arising  from  working 
steam  expansively,  262. 

Index  of  logarithms,  334. 

Indicator,  264,  265. 

Indicator,  the  amount  of  the  effective  power 
of  steam  by,  266. 

Induction  ports,  171. 

Inelastic  bodies,  449. 

Influence  of  pressure,  velocity,  width  of  fel- 
loes, and  diameter  of  wheels,  309. 

Initial  plane,  456,  480,  500. 

Initial  velocity  with  a  free  descent,  388. 

Injection  pipe,  150. 

Inside  discharging  turbine,  330. 

Integer,  10.  * 

Integers,  to  find  the  square  root  of,  125. 

Interest,  simple,  42. 

Interest,  compound,  43. 

Involution,  28. 

Involution,  or  the  raising  of  powers  by  loga- 
rithms, 338. 

Irregular  polygons,  54. 

Iron,  forged  and  wrought,  272. 

Iron  plates,  403. 

Iron,  properties  of,  175. 


Iron,  strength  of,  173. 

Iron,  taper  and  parallel,  angle  and  T,  rail- 
way and  sash,  408,  411. 

JET,  specific  gravity  of,  394. 

Journal  of  cross-head,  to  find  diameter  of.  139. 

Journal   of  cross-head,  to  find  the   length 

of,  139. 
Journal,  the  mean  centre,  to  find  the  diameter 

of,  143. 

Journal,  strain  of,  252. 
Journals  for  air-pump  cross-head,  145. 
Journals  for  shafts  of  various  diameters,  287. 
Julian  year,  357. 
Juste  Byrge,  the  inventor  of  logarithms,  133. 

KANE,  Fitzgerald,  269.   , 
Keel  and  keelson,  433  to  500. 
Kilometre,  5. 
Kilogramme,  6. 
Knots,  nodes,  <fcc.,  412. 

LATHE  spindle  wheel,  435. 

Laying  off"  of  angles  by  compasses  only,  384. 

Leg  of  a  spherical  triangle,  to  find,  377. 

Length  of  crank  pin  of  locomotive,  170. 

Length  of  paddle-shaft  journal,  138. 

Length  of  stroke,  227,  251. 

Lengths  that  may  be  given  to  stroke  of  the 

valve,  229. 

Lengths  of  circular  arcs,  68. 
Lever,  426. 

Light  displacement,  459. 
Line  of  direction,  390. 
Link  next  the  radius  bar,  242. 
Living  forces,  or  the  principle  of  vis  viva,270. 
Load  immersion,  456,  457. 
Load-water  line,  456,  478. 
Locomotive  engine,  parts  of  the  cylinder,  171. 
Locomotive  engine,  diameter  of  the  outside 

bearings  for,  163. 
Locomotive   engine,  dimensions   of  several 

moving  parts,  171. 
Locomotive  engine,  dimensions   of  several 

pipes,  171. 

Locomotive  engine,  parts  of  the  boiler,  171. 
Locomotive  engine,  tender  tank,  92. 
Locomotive  and  other  engines,  233. 
Logarithmic  calculations,  376. 
Logarithmic    calculations    of  the    force    of 

steam,  190  to  193. 
Logarithmic  sines,  tangents,  and  secants  for 

every  minute  in  the  quadrant,  540,  576. 
Logarithms  applied  to  angular  magnitudes, 

359. 

Logarithms,  hyperbolic,  130. 
Logarithms  of  the  natural  numbers  from  1 

to  100000  by  the  help  of  differences,  503 

to  540. 

Logarithms,  the  application  of,  334. 
Long  measure,  7. 
Longitudinal  distance  of  the  centre  of  gravity 

of  displacement,  470,  500. 
Loss  of  force  by  the  decrease  of  temperature 

in  the  steam  pipes,  221. 
Low  pressure  engines,  243. 
Lunes,  54. 

MACHINERY,  elements  of,  425. 
Machinery   worked  by  hydraulic   pressure, 
330. 


INDEX. 


587 


Major  and  minor  diameters  of  cross-head, 

253. 

Main  beam  at  centre,  249. 
Malleable  iron,  396. 
Marble,  288. 
Marine  boilers,  217. 
Mass,  267. 

Mass,  gravity,  and  weight,  386. 
Mass  of  a  body,  to  find,  when  the  weight  is 

given,  389. 
Materials  employed  in  the  construction  of 

machines,  267. 
Materials,  their  properties,  torsion,  deflexion, 

Ac.,  267. 

Maximum  accelerating  force,  421. 
Maximum   velocity   and    power    of   water 

wheels,  443. 

Measures  and  weights,  5. 
Measurement  of  angular  magnitudes.  374. 
Measurement  of  angles  by  compasses  only, 

382. 

Mechanical  effect,  417. 
Mechanical  powers,  422. 
Mechanical  power  of  steam,  261. 
Mensuration  of  solids,  79. 
Mensuration  of  timber,  93. 
Mensuration  of  superficies,  45. 
Mercury,  density  of,  350. 
Mercury,  to  calculate  the  force  of  steam  in 

inches  of,  201. 
Method  to  calculate  the  logarithm  of  any 

given  number,  340. 
Metacentre,  482.         4 
Metre,  5. 
Midship,  or  greatest  transverse  section,  460, 

487. 

Millboard,  405. 
Millstones,  445. 
Millstones,  strength  of,  451. 
Modulus  of  elasticity,  278. 
Modulus  of  logarithms,  343. 
Modulus  of  torsion  and  of  rupture,  279. 
Moment  of  inertia,  412. 
Motion  of  elastic  fluids,  205. 
Motion  of  steam  in  an  engine,  206. 
Multiplication  of  decimals,  23. 
Multiplication  of  fractions,  21. 
Multiplication  by  logarithms,  335. 
Musical  pi'oportion,  40. 

NATURAL  sines,  cosines,  tangents,  cotangents, 

secants,  and  cosecants,  to  every  degree  of 

the  quadrant,  411. 
Naval  architecture,  453. 
New  method  of  multiplication,  342. 
Nitrogen,  weight  off  356. 
Nominal  horse  power,  tables  of,  for  high  and 

low  pressure  engines,  243,  244. 
Notation  and  numeration,  10. 
Notation,  trigonometrical,  359. 
Number  corresponding  to  a  given  logarithm, 

351. 
Number  of   teeth,   or  the   pitch   of    small 

wheels,  435. 

Numbers,  fourth  and  fifth  powers  of,  129. 
Numbers,  logarithms  of,  540,  495. 
Numbers,  reciprocals  of,  73  to  78. 
Numbers,  squares,  cubes,  <fec.,  of,  100  to  116. 
Numeral   solution  of  the  several  cases   of 

trigonometry,  361. 
Nuts  and  bolts,  406. 


OAK,  Dantzic,  280. 

Obelisk,  to  find  the  height  of,  371. 

Oblique  triangles,  368. 

Observatory  at  Paris  g  =  9-80896  metres,346. 

O'Byrne's  turbine  tables,  331. 

Octagon,  48. 

Octaedron,  89. 

O'Neill's  experiments,  447. 

O'Neill's  rules  employed  in  the  art  of  ship- 
building, 454. 

Opium,  specific  gravity  of,  394. 

Orders  of  lever,  426. 

Ordinates  employed  in  the  art  of  ship-build- 
ing, 455,  456,  458. 

Orifices  and  tubes,  discharge  of  water  by,  312. 

Orifices,  rectangular,  314. 

Oscillation,  centre  of,  187,  391. 

Outside  bearings  of  crank  axle,  168. 

Outside  discharging  turbines,  331. 

Overshot  wheels,  329. 

Overshot  wheels,  maximum  velocity  of,  443. 

Ox-hide,  299. 

Oxygen,  214,  356. 

PADDLE-shaft  journal,  137,  251. 

Paraboloid,  88.  » 

Parabolic  conoid,  88. 

Parallel  angle  iron,  409. 

Parallel  motion,  242  to  246. 

Parallelogram  of  forces,  422. 

Parallelopipedon,  80. 

Partnership,  41. 

Partial  contraction  of  the  fluid  vein,  316. 

Passages,  area  of  steam,  220. 

Peclet's  expression  for  the  velocity  of  smoke 

in  chimneys,  213. 
Pendulums,  183,  391. 
Pendulum,  conical,  184. 
Pendulums,  vibrating  seconds  at  the  level  of 

the  sea  in  various  latitudes,  393. 
Percussion,  centre  of,  391. 
Periodic  time,  179. 

Permanent  weight  supported  by  beams,  284. 
Permutations  and  combinations,  44. 
Pillars,  strength  of,  293. 
Pinions  and  wheels  in   continuous   circular 

motion,  432. 
Pipes,    discharge    and    drainage   of   water 

through,  321,  322,  325. 
Pipes  of  cast  iron,  395. 
Pipes  for  marine  engines,  149. 
Piston,  251. 

Piston  of  steam  engine,  414. 
Piston  rod,  140, 171,  253. 
Piston  rod  of  air-pump,  146. 
Pitch  circle,  436. 
Pitch  of  teeth,  441. 
Pitch  of  wheels,  435,  439. 
Plane  triangles,  solution  of,  364,. 365. 
Plane  trigonometry,  359. 
Planks,  deals,  94. 
Polygons,  47,  48. 
Polygons,  irregular,  54. 
Port,  upper  and  lower,  229. 
Position,  double,  44. 
Position,  single,  43. 
Pound,  5. 

Power,  actual  and  nominal,  241. 
Power  and  properties  of  steam,  261. 
Power  that  a  cast-iron  wheel  is  capable  of 

transmitting,  442. 


588 


INDEX. 


Power  of  shafts,  294. 

Practical  application  of  the  rnechanica 
powers,  425. 

Practical  limit  to  expansion,  261. 

Practical  observations  on  steam  engines,  260 

Principle  of  virtual  velocities,  423. 

Prism,  80. 

Prismoid,  85. 

Properties  of  bodies,  401. 

Proportional  dimensions  of  nuts  and  bolts, 
406. 

Proportion,  14. 

Proportion,  musical,  40. 

Proportion  and  progression,  arithmetical,  35 
to  38. 

Proportion  and  progression,  geometrical,  38 
to  40. 

Proportion,  or  the  rule  of  three  by  loga- 
rithms, 338. 

Proportion  of  wheels  for  screw-cutting,  433. 

Proportions  of  boilers,  grates,  <fec.,  213. 

Proportions  of  the  lengths  of  circular  arcs,  68. 

Proportions  of  undershot  wheels,  328. 

Pulleys,  422,  427. 

Pump  and  pumping  engines,  446. 

Pumping  engines,  422. 

Pyramid,  82. 

Pyrometer,  63. 

QUADRANT,  359. 

Quadrant,  log.  sines,  cosines,  &c.,  for  every 
minute  in,  540,  576. 

Quadrant,  natural  sines  and  cosines  for 
every  degree  of,  411. 

Quadrant,  to  take  angles  with,  370. 

Quantities,  known  and  unknown,  134. 

Quantity  of  water  that  flows  through  a  cir- 
cular orifice,  313,  319. 

Quiescence,  friction  of,  299. 

RADIUS  bar,  242. 

Radius  bar,  length  of,  corrected,  248. 

Radius  of  the  earth  at  Philadelphia,  356. 

Radius  of  gyration,  412. 

Radius,  length  of,  in  degrees,  357. 

Rails,  temporary,  411. 

Railway  carriage,  268. 

Railway  iron,  410. 

Raising  of  powers  by  logarithms,  338. 

Reciprocals  of  numbers,  73  to  78. 

Recoil,  449. 

Rectangle,  rhombus,  rhomboides,  to  find 
the  areas  of,  45,  46. 

Reduction  of  fractions,  16,  17,  to  19,  20. 

Regnault's  experiments  on  oxygen,  &c.,  356. 

Regular  bodies,  90. 

Relative  capacities  of  the  two  bodies  under 
the  same  displacement,  456,  470. 

Relative  strength  of  materials  to  resist  tor- 
sion, 294. 

Revolving  shaft,  250. 

Riga  fir,  290. 

Right-angled  spherical  triangles,  374. 

Ring,  circular,  to  find  the  area  of,  53. 

Ring,  cylindrical,  90. 

Roads,  traction  of  carriages  on,  307. 

Rolled  iron,  395. 

Roman  notation,  11. 

Rope,  strength  of,  282. 

Ropes,  bands,  Ac.,  267. 

Ropes,  blocks,  pulleys,  428. 


Ropes,  stiffness  of,  resistance  of,  to  bendin 

302. 

Ropes,  tarred  and  dry,  304,  306. 
Rotative  engines,  260. 
Rotation,  moment  of,  414. 
Rotation  of  a  body  about  a  fixed  axis,  416. 
Rotations  of  millstones,  452. 
Round  and  rectangular  bars,  strength  of,  281. 
Round  bar-iron,  403. 
Round  steel  and  brass,  408. 
Rules  for  pumping  engines,  448. 
Rule  of  three,  13. 
Rule  of  three  by  logarithms,  338. 
Rule  of  three  in  fractions,  21. 
Rupture,  272. 

SAFETY  valves,  149, 150,  224. 

Sails  of  windmills,  332. 

Sash  iron,  410. 

Scales  of  chords,  how  to  construct,  360. 

Scale  of  displacement,  490. 

Scantling,  95. 

Screw  cutting  by  lathe,  433. 

Screw,  power  of,  430. 

Screw,  to  cut,  434. 

Sectional  area  measured,  456  to  468. 

Segments  of  circles,  64  to  67. 

Sheives,  cords,  blocks,  428. 

Ship-building  and  naval  architecture,  453. 

Sidereal  day,  9. 

Side  lever,  to  find  the  depth  across  the  centre 
of,  144. 

Side  rod,  246,  254. 

Side  rod  of  air-pump,  146. 

Sines,  cosines,  &c.,  411. 

Sines,  tangents  and  secants,  359. 

Singular  phenomena,  237. 

Sleigh,  268. 

Slide  valve,  225. 

Slide  valve,  a  cursory  examination  of,  232. 

Slopes  li  to  1,  2  to  1,  and  1  to  1,  97. 

Sluice  board,  316. 

Smoke  and  heated  air  in  chimneys,  202. 

Solid  inches  in  a  solid  foot,  96. 

Solids,  mensuration  of,  79. 

Space  described  by  a  body  during  a  free  de- 
scent in  vacuo,  388. 

Specific  gravity,  386,  391. 

Sphere,  85. 

Spheres,  397  to  400.. 

Spheroid,  86,  87,  88. 

Spherical  trigonometry,  373. 

Spheroidal  condition  of  water  in  boilers,  236. 

Spindle  and  screw  wheels,  434. 

Square,  to  find  the  area  of,  45. 

Square  and  sheet  iron,  402. 

Squares  and  square  roots  of  numbers,  100 
to  116. 

Square  root,  30. 

Square  root  of  fractions  and  mixed  numbers, 
31. 

Square  measure,  6. 

Stability,  459. 

Stars,  apparent  motion  of,  353. 

Statical  moment,  417. 

Steam  engine,  135. 

Steam  dome,  171. 

Steam  passages,  220. 

Steam  pipes,  loss  of  force  in,  222. 

Steam  port,  147,  148. 

Steam  room,  259. 


INDEX. 


589 


Steam,  elastic  force  of,  188  to  202. 

Steam,  temperature  of,  pressure  of,  172. 

Steam,  volume  of,  202  to  206. 

Steam,  weight  of,  204. 

Steel,  408. 

Steel,  cast,  409. 

Stiffness  of  ropes,  302,  306. 

Stowage,  503. 

Stowing  the  hold  of  a  vessel,  453,  456. 

Strap  at  cutter,  141. 

Strap,  mean  thickness  of,  at  and  before  cut- 
ter, 143. 

Strength  of  bodies,  282. 

Strength  of  boilers,  218. 

Strength  of  materials,  173,  271. 

Strength  of  rods  when  the  strain  is  wholly 
tensile,  250. 

Strength  of  the  teeth  of  cast  iron  wheels,  437. 

c"-ids  of  lever,  143. 

l-wheel  and  pinion,  434. 

fc attraction  of  decimals,  23. 

Subtraction  of  fractions,  21.  \ 

TABLE  by  which  to  determine  the  number  of 
teeth  or  pitch  of  small  wheels,  435. 

Table  containing  the  circumferences,  squares, 
cubes,  and  areas  of  circles,  from  1  to  100, 
advancing  by  a  tenth,  57,  58,  59,  60  to 
63. 

Table  containing  the  weight  of  columns  of 
water,  each  one  foot  in  length,  in  "pounds 
avoirdupois,  401. 

Table  containing  the  weight  of  square  bar 
iron,  402. 

Table  containing  the  surface  and  solidity  of 
spheres,  together  with  the  edge  of  equal 
cubes,  the  length  of  equal  cylinders,  and 
weight  of  water  in  avoirdupois  pounds, 
397. 

Table  containing  the  weight  of  flat  bar  iron, 
400. 

Table  containing  the  specific  gravities  and 
other  properties  of  bodies;  water  the  stand- 
ard of  comparison,  401. 

Table  containing  the  weight  of  round  bar 
iron,  403. 

Table  containing  the  weights  of  cast  iron 
pipes,  404. 

Table  containing  the  weight  of  solid  cylin- 
ders of  cast  iron,  404. 

Table  containing  the  weight  of  a  square  foot 
of  copper  and  lead,  405. 

Table  for  finding  the  weight  of  malleable 
iron,  copper,  and  lead,  405. 

Table  for  finding  the  radius  of  a  wheel  when 
the  pitch  is  given,  or  the  pitch  when  the  ra- 
dius is  given,  for  any  number  of  teeth,  439. 

Table  for  the  general  construction  of  tooth 
wheels,  442. 

1'able  for  breast  wheels,  329. 

Table  of  polygons,  48. 

Table  of  decimal  approximations  for  facili- 
tating calculations,  55. 

Table  of  decimal  equivalents,  56. 

Table  of  the  areas  of  the  segments  and  zones 
of  a  circle  of  which  the  diameter  is  unity, 
64,  65,  66,  67. 

Table  of  the  proportions  of  the  lengths  of 
semi-elliptic  arcs,  69,  70,  72. 

Table  of  flat  or  board  measure,  93. 

Table  of  solid  timber  measure,  94. 


Table  of  reciprocals  of  numbers,  or  of  the 
decimal  fractions  corresponding  to  com- 
mon fractions,  71  to  77,  78. 

Table  of  weights  and  values  in  decimal 
parts,  79. 

Table  of  regular  bodies,  90. 

Table  of  the  cohesive  power  of  bodies,  175. 

Table  of  hyperbolic  logarithms,  130  to  133. 

Table  of  the  pressure  of  steam,  in  inches  of 
mercury  at  different  temperatures,  172. 

Table  of  the  temperature  of  steam  at  differ- 
ent pressures,  in  atmospheres,  172. 

Table  of  the  expansion  of  air  by  heat,  173. 

Table  of  the  strength  of  iron,  173. 

Table  of  the  superficial  and  solid  content  of 
spheres,  96. 

Table  of  solid  inches  in  a  solid  foot,  96. 

Table  of  squares,  cubes,  square  and  cube 
roots,  of  numbers,  100,  101,  116,  125. 

Table  of  cover  on  the  exhausting  side  of  the 
valve  in  parts  of  the  stroke  and  distance 
of  piston  from  the  end  of  its  stroke,  231. 

Table  of  the  proportions  of  the  lengths  of 
circular  arcs,  68. 

Table  of  the  fourth  and  fifth  power  of  num- 
bers, 129. 

Table  of  the  properties  of  different  boil- 
ers, 215. 

Table  of  the  economical  effects  of  expan- 
sion, 216. 

Table  of  the  comparative  evaporative  power 
of  different  kinds  of  coal,  218. 

Table  of  the  cohesive  strength  of  iron  boiler 
plate  at  different  temperatures,  219. 

Table  of  diminution  of  strength  of  copper 
boilers,  219. 

Table  of  expanded  steam,  239. 

Table  of  the  proportionate  length  of  bearings, 
or  journals  for  shafts  of  various  diameters, 
287. 

Table  of  tenacities,  resistances  to  compres- 
sion and  other  properties  of  materials, 

OQO 
ZOO. 

Table  of  the  strength  of  ropes  and  chains, 
288. 

Table  of  the  strength  of  alloys,  289. 

Table  of  data  of  timber,  289. 

Table  of  the  properties  of  steam,  261. 

Table  of  the  mechanical  properties  of  steam, 
263. 

Table  of  the  cohesive  strength  of  bodies,  281. 

Table  of  the  strength  of  common  bodies,  283. 

Table  of  torsion  and  twisting  of  common  ma- 
terials, 286. 

Table  of  the  length  of  circular  arcs,  radius 
being  unity,  63. 

Table  of  experiments  on  iron  boiler  plate  at 
high  temperature,  220. 

Table  of  the  absolute  weight  of  cylindrical 
columns;  274. 

Table  of  flanges  of  girders,  276. 

Table  of  mean  pressure  of  steam  at  different 
densities  and  rates  of  expansion,  239. 

Table  of  nominal  horse  power  of  high  pres- 
sure engines,  244. 

Table  of  nominal  horse  power  of  low  pres- 
sure engines,  243. 

Table  of  dimensions  of  cylindrical  columns 
of  cast  iron  to  sustain  a  given  load  with 
safety,  293. 

Table  of  strength  of  columns,  294. 


590 


INDEX. 


Table  of  comparative  torsion,  294. 

Table  of  the  depths  of  square  beams  to  sup- 
port from  1  cwt.  to  14  tons,  295,  296. 

Table  of  the  results  of  experiments  on  fric- 
tions, with  unguents  interposed,  299,  300. 

Table  of  the  results  of  experiments  on  the 
gudgeons  or  axle-ends  in  motion  upon  their 
bearings,  301. 

Table  of  friction,  continued  to  abrasion,  301. 

Table  of  friction  of  steam  engines  of  differ- 
ent modifications,  302. 

Table  of  tarred  ropes,  303. 

Table  of  white  ropes,  305. 

Table  of  dry  and  tarred  ropes,  306. 

Table  of  the  pressure  and  traction  of  car- 
riages, 308. 

Table  of  traction  of  wheels,  309. 

Table  of  the  ratio  of  traction  to  the  load, 
310. 

Table  of  the  coefficients  of  the  efflux  through 
rectangular  orifices  in  a  thin  vertical  plate, 
315. 

Table  of  the  coefficients  of  efflux,  315. 

Table  of  comparison  of  the  theoretical  with 
the  real  discharges  from  an  orifice,  317. 

Table  of  discharge  of  tubes  of  different  en- 
largements, 322. 

Table  of  the  comparison  of  discharge  by  pipes 
of  different  lengths,  323. 

Table  of  the  comparison  of  discharge  by  ad- 
ditional tubes,  323. 

Table  of  the  friction  of  fluids,  325. 

Table  of  discharges  of  a  6-inch  pipe  at  seve- 
ral inclinations,  326. 

Table  of  the  velocity  of  windmill  sails,  333. 

Table  of  outside  discharging  turbine,  331. 

Table  of  inward  discharging  turbines,  332. 

Table  of  peculiar  logai-ithms,  340. 

Table  of  useful  logarithms,  345. 

Table  of  the  specific  gravity  of  various  sub- 
stances, $94. 

Table  of  the  weight  of  a  foot  in  length  of  flat 
and  rolled  iron,  395. 

Table  of  the  weight  of  cast  iron  pipes,  395. 

Table  of  the  weight  of  one  foot  in  length  of 
malleable  iron,  396. 

Table  of  comparison,  396. 

Table  of  the  weight  of  a  square  foot  of  sheet 
iron,  402. 

Table  of  the  weight  of  a  square  foot  of  boiler 
plate  from  £  of  an  inch  to  1  inch  thick,  403. 

Table  of  the  weights  of  cast  iron  plates,  403. 

Table  of  the  weight  of  mill-board,  405. 

Table  of  the  weight  of  wrought  iron  bars,  406. 

Table  of  the  proportional  dimensions  of  nuts 
and  bolts,  406. 

Table  of  the  specific  gravity  of  water  at  dif- 
ferent temperatures,  406. 

Table  of  the  weight  of  cast  iron  balls,  407. 

Table  of  the  weight  of  flat  bar  iron,  407. 

Table  of  the  weight  of  square  and  round 
brass,  408. 

Table  of  taper  T  iron,  410. 

Table  of  sash  iron,  410. 

Table  of  rails  of  equal  top  and  bottom,  410. 

Table  of  temporary  rails,*v411. 

Table  of  natural  sines,  cosines,  tangents,  co- 
tangents, secants,  and  cosecants,  to  every 
degree  of  the  quadrant,  411. 

Table  of  inclined  planes,  showing  the  ascent 
or  descent  the  yard,  430. 


Table  of  the  weight  of  round  steel,  408. 

Table  of  parallel  angle  iron  of  equal  sides,  408. 

Table  of  parallel  angle  iron  of  unequal  sides, 
409. 

Table  of  taper  angle  iron  of  equal  sides,  409. 

Table  of  parallel  T  iron  of  unequal  width  and 
depth,  409. 

Table  of  change  wheels  for  screw-cutting. 
435. 

Table  of  the  diameters  of  wheels  at  their 
pitch  circle,  to  contain  a  required  number 
of  teeth,  436. 

Table  of  the  angle-  of  windmill  sails,  445. 

Table  of  the  logarithms  of  the  natural  num- 
bers, from  1  to  100000,  by  the  help  of  dif- 
ferences, 495  to  540. 

Table  of  log.  sines,  cosines,  tangents,  cotan- 
gents, secants  and  cosecants,  for  every  de- 
gree and  minute  in  the  quadrant,  540  to 
576. 

Table  of  the  strength  of  the  teeth  of  cast  iron 
wheels  at  a  given  velocity,  437. 

Table  of  approved  proportions  for  wheels 
with  flat  arms,  441. 

Table  showing  the  cover  required  on  the 
steam  side  of  the  valve  to  cut  the  steam  off 
at  any  part  of  the  stroke,  228. 

Table  showing  the  cover  required,  227. 

Table  showing  the  resistance  opposed  to 
the  motion  of  carriages  on  different  incli- 
nations of  ascending  or  descending.planes, 
429. 

Table  showing  the  number  of  linear  feet  of 
scantling  of  various  dimensions  which  are 
equal  to  a  cubic  foot,  95. 

Table  showing  the  weight  or  pressure  a  beam 
of  cast  iron  will  sustain  without  destroying 
its  elastic  force,  292. 

Table  showing  the  circumference  of  rope 
equal  to  a  chain,  282. 

Table  to  correct  parallel  motion  links,  248. 

Table  of  parallel  T  iron  of  equal  depth  and 
width,  410. 

Tables  of  cuttings  and  embankments,  slopes, 
1  to  1  j  H  to  1  j  and  2  to  1,  97. 

Tables  of  the  heights  corresponding  to  differ- 
ent velocities,  389. 

Tables  of  the  mechanical  properties  of  the 
materials  most  commonly  employed  in  the 
construction  of  machines  and  framings, 
280. 

Tangents,  360. 

Tangents  and  secants,  to  compute,  362. 

Taper  angle  iron,  410. 

Teeth  of  wheels  in  continuous  circular  motion, 
432. 

Teeth  of  wheels,  422,  436. 

Temperature  of  steam,  172. 

Temperature  and  elastic  force  of  steam,  188. 

Tension  of  chain-bridge,  414. 

Tetraedron,  89. 

Threshing  machines,  445. 

Throttling  the  steam,  234. 

Timber  measure,  93. 

Timber,  to  measure  round,  95. 

Time,  7. 

Tonnage  of  ships,  461  to  494. 

Torsion,  279. 

Torsion  and  twisting,  286. 

Traction  of  carriages,  307. 

Transverse  strength  of  bodies,  282. 


INDEX. 


591 


Transverse  strain,  278. 

Transverse  strain,  time  weight,  273. 

Trapezium,  47. 

Trapezoid,  47. 

Triangle,  to  find  the  area  of,  46,  47. 

Trigonometry,  359. 

Trigonometry,  spherical,  373. 

Troy  weight,  7. 

Trussed  beams,  291. 

Tubes,  discharge  of  water  through,  312. 

Tubular  boilers,  257. 

Turbine  water-wheels,  330. 

ULTIMATE  pressure  of  expanded  steam,  236. 

Undecagon,  47. 

Undershot  wheels,  327,  443. 

Unguents,  299. 

Ungulas,  cylindrical,  81. 

Ungulas,  conical,  83,  84. 

Unit  of  length,  5. 

Unit  of  weight,  5. 

Unit  of  dry  capacity,  5. 

Units  of  liquids,  5. 

Units  of  work,  269,  297,  414,  4lo. 

Universal  pitch  table,  442. 

Upper  steam  port,  229. 

Useful  formula,  271. 

Use  of  the  table  of  squares,  cubes,  &c.,  127. 

VACUUM,  perfect  one,  235. 
Vacuum  below  the  piston,  251. 
Vacuo,  bodies  falling  freely  in,  388. 
Valves,  different  arrangements  of,  233. 
Valve,  length  of  stroke  of,  in  inches,  228. 
Valve  shaft,  147. 
Valve,  safety,  224. 
Valve,  slide,  225. 
Valve  spindle,  171. 
Vapour  in  the  cylinder,  229. 
Vein,  contraction  of  fluid,  330. 
Velocity,  force,  and  work  done,  267. 
Velocity  of  steam  rushing  into  a  vacuum,  207. 
Velocity  of  smoke  in  chimneys,  209,  213. 
Velocity  of  piston  of  steam  engine,  266. 
Velocity  of  threshing  machines,  millstones, 

boring,  <fcc.,  445. 

Velocity  of  wheels  on  ordinary  roads,  307. 
Venturi,  experiments  of,  on  the  discharge  of 

fluids,  421. 

Versed  sine,  tabular,  52. 
Versed  sine  of  parallel  motion   '-M:. 
Yersed  sine,  359. 


Vertical  sectional  areas,  454. 

Virtual  velocities,  424. 

Vis  viva,  principle  of,  calculations  on,  276, 

388. 

Volume  of  a  ship  immersed,  456. 
Volume  of  steam  in  a  cubic  foot  of  water, 

202,  205. 

WATER,  modulus  of  elasticity  of,  190. 
Water  level,  214. 
Water,  feed  and  condensing,  223. 
Water,  spheroidal  condition  of,  inboilers,236. 
Water  in  boiler,  and  water  level,  358. 
Water,  discharge  of,  through  different  orifi- 
ces, 312,  318. 
Water  wheels,  327. 

Water  wheels,  maximum  velocity  of,  443. 
Web  of  crank  at  paddle  shaft  centre,  136. 
Web  of  cross-head  at  middle,  139. 
Web  of  crank  at  pin  centre,  142. 
Web  at  paddle  centre,  252. 
Web  of  cross-head  at  journal,  140. 
Web  of  air-pump  cross-head,  145. 
Wedge,  85. 

Wedge  and  screw,  430. 
Weights  and  measures,  5. 
Weights,  values  of,  in  decimal  parts,  79. 
Weight,  mass,  gravity,  386 . 
Weirs,  and  rectangular  apertures,  314,  323. 
Wheel  and  axle,  417. 
Wheel  and  pinion,  427. 
Wheels,  drums,  pulleys,  438. 
Windmills,  332. 
Wine  measure,  8. 
Woods,  280. 

Woods,  specific  gravity,  394. 
Work  done,  weight,  267. 
Wrought  iron  bars,  406. 

YARD,  5. 

Yacht,  admeasurement  of,  466,  470. 

Yarns  of  ropes,  303. 

Yellow  brass,  281. 

Yew,  280. 

ZINC,  280. 

Zinc,  sheet,  288. 

Zone,  spherical,  86. 

Zone,  to  find  the  area  of  a  circular,  53. 

Zones  of  circles,  to  find  the  areas  of,  64,  65, 


THE  END. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


-V  1942 


LD  21-100m-7,'40 (6936s) 


7     XT/ 


